Primes k-tuple (2)

K-Tuple

Competition #5

This competition is organized by Макарова Наталия (Natalia Makarova)

Introduction

Basic definitions of k-tuples primes can be found in [1] and [3].

Definition 1:

A prime k-tuple is a finite collection of values (p + a1, p + a2, p + a3, …, p + ak), where p, p + a1, p + a2, p + a3, …, p + ak are prime numbers, (a1, a2, a3, …, ak) are pattern. Typically the first value in the pattern is 0 and the rest are distinct positive even numbers.

Definition 2:

The diameter d of k-tuple is the difference of its largest and smallest elements.

We are considering k-tuples of consecutive primes, so
p + a1, p + a2, p + a3, …, p + ak
are sequential prime numbers.
We consider symmetric k-tuples.
See Definition 2 and Definition 3 in [3].

Definition 3:

k-tuple (p + a1, p + a2, p + a3, …, p + a [k / 2], p + a [k / 2+1], …, p + a [k-2], p + a [k-1], p + ak) for k even, is called symmetric, if the following condition is satisfied:
a1 + ak = a2 + a[k-1] = a3 + a[k-2] = … = a[k/2] + a[k/2+1]

Definition 4:

k-tuple (p + a1, p + a2, p + a3, …, p + a [(k-1) / 2], p + a [(k-1) / 2 + 1], p + a [(k-1) / 2 + 2], …, p + a [k-2], p + a [k-1], p + ak) for k odd called symmetric, if the following condition is satisfied:
a1 + ak = a2 + a[k-1] = a3 +a [k-2] =…= a[(k-1)/2] + a[(k-1)/2+2] = 2 a[(k-1)/2+1]

We write the k-tuple pattern this way

0, p_2, p_3, …, p_k

where p_k are even numbers.

For example, a pattern for 17-tuple

0, 6, 24, 36, 66, 84, 90, 114, 120, 126, 150, 156, 174, 204, 216, 234, 240

We write the k-tuple this way

X: 0, p_2, p_3, …, p_k

where X is a prime number.

For example, 17-tuple (author J. Wroblewski)

1006882292528806742267: 0, 6, 24, 36, 66, 84, 90, 114, 120, 126, 150, 156, 174, 204, 216, 234, 240

The most difficult task is to find the minimum k-tuple with the minimum diameter d.
In such a k-tuple, for the minimum value of d, the value of X is the minimum.

For example, a minimum 17-tuple with a minimum diameter of 240 (author J. Wroblewski)

258406392900394343851: 0, 12, 30, 42, 60, 72, 78, 102, 120, 138, 162, 168, 180, 198, 210, 228, 240

d = 240 (minimum)
X = 258406392900394343851 (minimum)

For the competition there are 7 tasks that you can compete.

Tasks

Task #1

Search 17-tuple with pattern

0, 6, 24, 36, 66, 84, 90, 114, 120, 126, 150, 156, 174, 204, 216, 234, 240

Currently there are 8 known solutions (authors J. Wroblewski and D. Petukhov)

1006882292528806742267: 0 6 24 36 66 84 90 114 120 126 150 156 174 204 216 234 240
3954328349097827424397: 0 6 24 36 66 84 90 114 120 126 150 156 174 204 216 234 240
4896552110116770789773: 0 6 24 36 66 84 90 114 120 126 150 156 174 204 216 234 240
6751407944109046348063: 0 6 24 36 66 84 90 114 120 126 150 156 174 204 216 234 240
7768326730875185894807: 0 6 24 36 66 84 90 114 120 126 150 156 174 204 216 234 240
19252814175273852997757: 0 6 24 36 66 84 90 114 120 126 150 156 174 204 216 234 240
154787380396512840656507: 0 6 24 36 66 84 90 114 120 126 150 156 174 204 216 234 240
901985248981556228168767: 0 6 24 36 66 84 90 114 120 126 150 156 174 204 216 234 240

We do not guarantee that there are no missing solutions in this list.
It is required to find missing solutions, if they exist.
And you also need to continue the list of solutions, that is, find solutions for X > 901985248981556228168767.

See [7].

Task #2

Currently two 19-tuples are known

6919940122097246303: 0, 48, 78, 138, 198, 204, 210, 264, 288, 294, 300, 324, 378, 384, 390, 450, 510, 540, 588
7325015925425379457: 0, 6, 30, 90, 126, 132, 150, 162, 216, 246, 276, 330, 342, 360, 366, 402, 462, 486, 492

Solutions found in the BOINC project SPT. [9]

It is not known whether the first solution is minimal.
We need to find the minimum solution, if it exists.
The diameter can be any.
In other words, we need to find a solution

Х: 0, p_2, p_3, …, p_17, p_18, p_19

for X < 6919940122097246303
if such a solution exists.

See [6].

Task #3

You need to find a 19-tuple with a minimum diameter d = 252

Х: 0, 6, 12, 30, 42, 72, 90, 96, 120, 126, 132, 156, 162, 180, 210, 222, 240, 246, 252

If you guarantee that X is minimal, that’s good, but that condition doesn’t have to be true.
So far we do not know any solution for this task.

Task #1 can help you complete this task.

See [7] and [10].

Task #4

Two 18-tuples with a minimum diameter d = 82 are known (author J. Wroblewski)

824871967574850703732309: 0, 4, 10, 12, 18, 22, 28, 30, 40, 42, 52, 54, 60, 64, 70, 72, 78, 82 
2124773992554613163708029: 0, 4, 10, 12, 18, 22, 28, 30, 40, 42, 52, 54, 60, 64, 70, 72, 78, 82

It is not known whether the first solution is minimal.
Need to find 18-tuple

Х: 0, 4, 10, 12, 18, 22, 28, 30, 40, 42, 52, 54, 60, 64, 70, 72, 78, 82

for X < 824871967574850703732309, if such a solution exists.

See [11] and [12].

Task #5

Known 20-tuple with a minimum diameter d = 94 (authors N. Makarova and J. Wroblewski)

824871967574850703732303: 0, 6, 10, 16, 18, 24, 28, 34, 36, 46, 48, 58, 60, 66, 70, 76, 78, 84, 88, 94

It is unknown whether this solution is minimal.
We need to find the minimum solution, if it exists.

Theoretical patterns for 20-tuple with minimum diameter

0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 60, 64, 66, 70, 76, 78, 84, 88, 90, 94
0, 4, 6, 10, 16, 18, 24, 28, 34, 36, 58, 60, 66, 70, 76, 78, 84, 88, 90, 94
0, 4, 6, 10, 16, 18, 24, 28, 36, 46, 48, 58, 66, 70, 76, 78, 84, 88, 90, 94
0, 4, 6, 10, 16, 18, 24, 30, 34, 46, 48, 60, 64, 70, 76, 78, 84, 88, 90, 94
0, 4, 6, 10, 16, 18, 24, 34, 36, 46, 48, 58, 60, 70, 76, 78, 84, 88, 90, 94
0, 6, 10, 16, 18, 24, 28, 34, 36, 46, 48, 58, 60, 66, 70, 76, 78, 84, 88, 94

See [12], [13] and [15].

Task #6

Currently two 26-tuples are known

5179852391836338871: 0, 12, 18, 28, 46, 76, 78, 120, 186, 210, 226, 232, 238, 300, 306, 312, 328, 352, 418, 460, 462, 492, 510, 520, 526, 538
7331618973503379271: 0, 22, 30, 36, 40, 52, 58, 72, 120, 160, 190, 192, 220, 348, 376, 378, 408, 448, 496, 510, 516, 528, 532, 538, 546, 568

Solutions found in the BOINC project SPT. [9]

It is not known whether the first solution is minimal.
Need to find 26-tuple

Х: 0, p_2, p_3, …, p_24, p_25, p_26

for X < 5179852391836338871, if such a solution exists.

See [9].

Task #7

Need to find 21-tuple

Х: 0, p_2, p_3, …, p_19, p_20, p_21

for any X and for any diameter d.

For minimum d = 324 there are two theoretical patterns

0, 12, 30, 42, 54, 60, 72, 84, 114, 120, 162, 204, 210, 240, 252, 264, 270, 282, 294, 312, 324
0, 12, 30, 42, 54, 60, 84, 114, 120, 144, 162, 180, 204, 210, 240, 264, 270, 282, 294, 312, 324

See [14] and [15].

Rules

You can complete one or more of the seven suggested tasks.
You can use the algorithms and programs that you will find at the links below.

You are allowed to use any computer technology, including clusters and supercomputers.

When you send a solution in the submit entry page, to help as in verify it, please enter a tuple in this format:

T# 
X: 0, p_2, p_3, …, p_k

Where T is the task number  and the rest is the k-tuple in standard format in a new line.

Organizational matters

The competition begins on January 4, 2024 and will last six months (ending on July 4, 2024).
Solutions found during the competition will not be published until the end of the competition.

Solutions must be entered in the specified format only (see Rules).
Solutions will be checked by the competition organizers.

The winner of the competition will receive a prize of 10,000 RUR.
For a foreign winner, the amount will be converted into US dollars at the Russian exchange rate on the day the competition ends.

The winner will be determined by the competition organizers based on the totality of submitted solutions.

Organizers

Natalia Makarova
contact [email protected]

Stefano Tognon
contact [email protected]

Links

  1. Prime k-tuple
    https://en.wikipedia.org/wiki/Prime_k-tuple
  2. Problem 60. Symmetric primes on each side.
    http://www.primepuzzles.net/problems/prob_060.htm
  3. Problem 62. Symmetric k-tuples of consecutive primes
    http://www.primepuzzles.net/problems/prob_062.htm
  4. Smallest prime starting a sequence of 2n consecutive odd primes with symmetrical gaps about the center.
    https://oeis.org/A055382
  5. Symmetric tuples of sequential primes in OEIS
    https://boinc.progger.info/odlk/forum_thread.php?id=259
  6. Central prime p in the smallest (2n+1)-tuple of consecutive primes that are symmetric with respect to p.
    https://oeis.org/A055380
  7. Development of a new algorithm
    https://boinc.progger.info/odlk/forum_thread.php?id=268
  8. Symmetric tuples of sequential primes
    https://dxdy.ru/topic100750.html
  9. BOINC-project SPT
    https://boinc.termit.me/adsl/
  10. Symmetric tuple of length 19 with minimum diameter of consecutive primes
    https://boinc.progger.info/odlk/forum_thread.php?id=269
  11. Minimization problem 18-tuple with minimum diameter
    https://boinc.progger.info/odlk/forum_thread.php?id=266
  12. Smallest prime starting a (nonsingular) symmetric n-tuplet of the shortest span (=A266511(n)).
    https://oeis.org/A266512
  13. Minimization problem 20-tuple with minimum diameter
    https://boinc.progger.info/odlk/forum_thread.php?id=267
  14. Symmetric tuple of length 21 with minimum diameter of consecutive primes
    https://boinc.progger.info/odlk/forum_thread.php?id=270
  15. Natalia Makarova and Vladimir Chirkov, Theoretical patterns with a minimal diameter for a(2) – a(50)
    https://oeis.org/A266512/a266512_1.txt

Result for K-Tuples of Primes

K-Tuple, Result, ,

The competition K-Tuples of Primes is over.

Thanks to all people that have partecipate.

The winner is Jarek (see Ranking)

The result is in this table:

Task
K-tuple
Diameter
Prime
Magic(*)
Name
1
17
240
1006882292528806742267
0
Jarek
0,6,24,36,66,84,90,114,120,126,150,156,174,204,216,234,240
1
17
348
11954696436290948869
0
Jarek
0,18,60,78,90,138,144,168,174,180,204,210,258,270,288,330,348
1
17
360
12196464604998841777
0
Jarek
0,36,84,96,114,120,126,150,180,210,234,240,246,264,276,324,360
1
17
264
1338977422865229706499
0
Jarek
0,12,24,30,42,54,84,90,132,174,180,210,222,234,240,252,264
1
17
360
14271237683005753507
0
Jarek
0,24,30,84,96,114,150,156,180,204,210,246,264,276,330,336,360
1
17
348
17667344133365404873
0
Jarek
0,30,48,84,90,108,114,150,174,198,234,240,258,264,300,318,348
1
17
420
18462005826764715791
0
Jarek
0,78,90,102,120,162,168,192,210,228,252,258,300,318,330,342,420
1
17
240
19252814175273852997757
0
Jarek
0,6,24,36,66,84,90,114,120,126,150,156,174,204,216,234,240
1
17
240
20278587540464136529199
0
Jarek
0,12,30,42,60,72,78,102,120,138,162,168,180,198,210,228,240
1
17
264
2035559077035293441299
0
Jarek
0,12,24,42,54,84,90,114,132,150,174,180,210,222,240,252,264
1
17
240
24300494153317939112651
0
Jarek
0,12,18,30,42,72,78,102,120,138,162,168,198,210,222,228,240
1
17
240
25651315879379564172971
0
Jarek
0,12,18,30,42,72,78,102,120,138,162,168,198,210,222,228,240
1
17
240
258406392900394343851
0
Jarek
0,12,30,42,60,72,78,102,120,138,162,168,180,198,210,228,240
1
17
288
311634572279873026493
0
Jarek
0,18,54,60,78,84,120,138,144,150,168,204,210,228,234,270,288
1
17
240
32686971428909208943211
0
Jarek
0,12,30,42,60,72,78,102,120,138,162,168,180,198,210,228,240
1
17
264
384703558068522780559
0
Jarek
0,24,30,42,72,84,90,114,132,150,174,180,192,222,234,240,264
1
17
240
3954328349097827424397
0
Jarek
0,6,24,36,66,84,90,114,120,126,150,156,174,204,216,234,240
1
17
252
401276622469261903031
0
Jarek
0,6,12,30,72,90,96,120,126,132,156,162,180,222,240,246,252
1
17
324
443707110791502007579
0
Jarek
0,42,72,84,114,120,132,150,162,174,192,204,210,240,252,282,324
1
17
240
4896552110116770789773
0
Jarek
0,6,24,36,66,84,90,114,120,126,150,156,174,204,216,234,240
1
17
288
535010601740877140023
0
Jarek
0,18,54,60,78,84,120,138,144,150,168,204,210,228,234,270,288
1
17
252
568398209014995678701
0
Jarek
0,6,12,30,42,72,90,96,126,156,162,180,210,222,240,246,252
1
17
240
6751407944109046348063
0
Jarek
0,6,24,36,66,84,90,114,120,126,150,156,174,204,216,234,240
1
17
312
6837359459759035391
0
Jarek
0,42,60,66,72,120,126,150,156,162,186,192,240,246,252,270,312
1
17
444
702939111495760681807
0
Jarek
0,90,102,132,144,174,180,210,222,234,264,270,300,312,342,354,444
1
17
252
752853880537802642981
0
Jarek
0,6,12,30,42,72,96,120,126,132,156,180,210,222,240,246,252
1
17
240
7768326730875185894807
0
Jarek
0,6,24,36,66,84,90,114,120,126,150,156,174,204,216,234,240
1
17
360
7902083290948579129
0
Jarek
0,12,18,102,132,138,150,168,180,192,210,222,228,258,342,348,360
1
17
360
8053379680763235601
0
Jarek
0,18,48,60,102,132,138,150,180,210,222,228,258,300,312,342,360
2
14
56
10421030292115097
0
Natalia Makarova
0, 2, 6, 12, 14, 20, 26, 30, 36, 42, 44, 50, 54, 56
2
17
240
258406392900394343851
0
Jarek
0,12,30,42,60,72,78,102,120,138,162,168,180,198,210,228,240
2
15
180
3112462738414697093
0
Jarek
0,6,24,30,54,66,84,90,96,114,126,150,156,174,180
2
12
46
41280160361347
0
Natalia Makarova
0, 4, 6, 10, 12, 22, 24, 34, 36, 40, 42, 46
2
20
94
824871967574850703732303
0
Jarek
0,6,10,16,18,24,28,34,36,46,48,58,60,66,70,76,78,84,88,94
2
20
94
824871967574850703732303
0
Natalia Makarova
0, 6, 10, 16, 18, 24, 28, 34, 36, 46, 48, 58, 60, 66, 70, 76, 78, 84, 88, 94
2
18
82
824871967574850703732309
0
Jarek
0,4,10,12,18,22,28,30,40,42,52,54,60,64,70,72,78,82
2
16
74
824871967574850703732313
0
Jarek
0,6,8,14,18,24,26,36,38,48,50,56,60,66,68,74
2
16
74
996689250471604163
0
Natalia Makarova
0, 6, 8, 14, 18, 24, 26, 36, 38, 48, 50, 56, 60, 66, 68, 74
3
16
94
320572022166380833
188
Jarek
0, 6, 10, 16, 18, 24, 28, 34, 60, 66, 70, 76, 78, 84, 88, 94
0, 88, 16, 84, 76, 24, 60, 28, 78, 10, 94, 6, 34, 66, 18, 70
3
16
94
320572022166380833
188
Natalia Makarova
0, 6, 10, 16, 18, 24, 28, 34, 60, 66, 70, 76, 78, 84, 88, 94
0, 88, 16, 84,76, 24, 60, 28, 78, 10, 94, 6, 34, 66, 18, 70
3
16
94
3848563151802671563
188
Jarek
0, 6, 10, 16, 18, 24, 28, 34, 60, 66, 70, 76, 78, 84, 88, 94
0,88,16,84,76,24,60,28,78,10,94,6,34,66,18,70
3
16
104
1798875821147433863
208
Jarek
0,6,14,20,24,30,38,44,60,66,74,80,84,90,98,104
0,98,20,90,80,30,60,38,84,14,104,6,44,66,24,74
3
16
122
1960984050584219159
244
Jarek
0,2,30,32,42,44,48,50,72,74,78,80,90,92,120,122
0,92,32,120,80,72,48,44,90,2,122,30,74,78,42,50
3
16
122
3919782822690082529
244
Jarek
0,12,20,32,42,48,54,60,62,68,74,80,90,102,110,122
0,110,32,102,80,54,48,62,90,20,122,12,74,60,42,68
3
16
134
2257802843933625383
268
Jarek
0,14,30,36,44,50,54,66,68,80,84,90,98,104,120,134
0,104,44,120,98,66,54,50,90,14,134,30,80,84,36,68
3
16
134
2618338000143236213
268
Jarek
0,6,8,14,30,36,38,44,90,96,98,104,120,126,128,134
0,128,14,126,104,36,90,38,120,8,134,6,44,96,30,98
3
16
136
3761006076207114997
272
Jarek
0,10,12,22,30,40,42,52,84,94,96,106,114,124,126,136
0,124,22,126,106,42,84,40,114,10,136,12,52,96,30,94
3
16
136
4363602266560571527
272
Jarek
0,10,12,22,42,52,54,64,72,82,84,94,114,124,126,136
0,124,22,126,94,54,72,52,114,10,136,12,64,84,42,82
3
16
136
6244416299036828467
272
Jarek
0,10,24,34,36,46,60,66,70,76,90,100,102,112,126,136
0,112,34,126,100,60,66,46,102,10,136,24,70,90,36,76
3
16
140
6691809871504172069
280
Jarek
0,20,30,42,48,50,62,68,72,78,90,92,98,110,120,140
0,110,50,120,98,72,48,62,90,20,140,30,92,78,42,68
3
16
146
196410946169488007
292
Jarek
0,20,24,42,44,60,62,66,80,84,86,102,104,122,126,146
0,122,44,126,104,66,60,62,102,20,146,24,86,84,42,80
3
16
146
2385895716615665027
292
Jarek
0,12,30,42,44,56,60,72,74,86,90,102,104,116,134,146
0,134,56,102,116,42,60,74,90,44,146,12,86,72,30,104
3
16
146
4018288550284158077
292
Jarek
0,2,12,14,42,44,54,56,90,92,102,104,132,134,144,146
0,134,14,144,104,54,90,44,132,2,146,12,56,102,42,92
3
16
146
5852971406868161687
292
Jarek
0,14,30,42,44,56,60,72,74,86,90,102,104,116,132,146
0,116,44,132,104,72,60,56,102,14,146,30,86,90,42,74
3
16
148
790313279999714941
296
Jarek
0,10,18,28,42,52,60,70,78,88,96,106,120,130,138,148
0,130,28,138,106,60,78,52,120,10,148,18,70,96,42,88
3
16
152
6118066623221589779
304
Jarek
0,2,30,32,42,44,72,74,78,80,108,110,120,122,150,152
0,122,32,150,110,72,78,44,120,2,152,30,74,108,42,80
3
16
154
3126035304223644013
308
Jarek
0,6,18,24,60,66,70,76,78,84,88,94,130,136,148,154
0,148,76,84,136,24,60,88,78,70,154,6,94,66,18,130
3
16
154
5908915623504697003
308
Jarek
0,6,30,36,48,54,70,76,78,84,100,106,118,124,148,154
0,148,76,84,124,36,48,100,78,70,154,6,106,54,30,118
3
16
158
256102275341453231
316
Jarek
0,18,20,38,42,60,62,78,80,96,98,116,120,138,140,158
0,140,38,138,116,60,78,62,120,20,158,18,80,96,42,98
3
16
158
2714164755369606551
316
Jarek
0,6,12,18,50,56,62,68,90,96,102,108,140,146,152,158
0,152,56,108,146,18,90,62,102,50,158,6,68,96,12,140
3
16
160
5848893302197701529
320
Jarek
0,12,30,42,48,60,70,78,82,90,100,112,118,130,148,160
0,148,82,90,130,42,48,100,78,70,160,12,112,60,30,118
3
16
160
64747964125739269
320
Jarek
0,12,30,42,58,60,70,72,88,90,100,102,118,130,148,160
0,148,70,102,130,42,60,88,90,58,160,12,100,72,30,118
3
16
164
1464641505981883193
328
Jarek
0,6,20,26,48,54,68,74,90,96,110,116,138,144,158,164
0,158,26,144,116,54,90,68,138,20,164,6,74,96,48,110
3
16
164
1960292815770022553
328
Jarek
0,14,30,36,44,50,66,80,84,98,114,120,128,134,150,164
0,134,44,150,128,66,84,50,120,14,164,30,80,114,36,98
3
16
164
274205528562749933
328
Jarek
0,6,30,36,44,50,74,80,84,90,114,120,128,134,158,164
0,158,50,120,134,36,84,74,114,44,164,6,80,90,30,128
3
16
164
2997980939880952853
328
Jarek
0,18,20,38,60,66,78,80,84,86,98,104,126,144,146,164
0,146,38,144,104,78,66,80,126,20,164,18,98,84,60,86
3
16
164
371373613280965583
328
Jarek
0,14,30,44,54,66,68,80,84,96,98,110,120,134,150,164
0,134,44,150,110,84,66,68,120,14,164,30,98,96,54,80
3
16
164
5526360461238138947
328
Jarek
0,12,20,32,42,54,62,74,90,102,110,122,132,144,152,164
0,152,32,144,122,54,90,62,132,20,164,12,74,102,42,110
3
16
164
6326577705299871353
328
Jarek
0,6,20,26,48,54,68,74,90,96,110,116,138,144,158,164
0,158,26,144,116,54,90,68,138,20,164,6,74,96,48,110
3
16
166
3616410488635421107
332
Jarek
0,10,12,22,42,52,54,64,102,112,114,124,144,154,156,166
0,154,22,156,124,54,102,52,144,10,166,12,64,114,42,112
3
16
166
6917484059804434777
332
Jarek
0,12,30,40,42,52,70,82,84,96,114,124,126,136,154,166
0,154,52,126,136,42,84,70,114,40,166,12,82,96,30,124
3
16
170
1038966937575833507
340
Jarek
0,14,30,36,44,50,66,80,90,104,120,126,134,140,156,170
0,140,44,156,134,66,90,50,126,14,170,30,80,120,36,104
3
16
170
1963958718695104277
340
Jarek
0,6,20,26,60,66,80,84,86,90,104,110,144,150,164,170
0,164,26,150,110,66,84,80,144,20,170,6,86,90,60,104
3
16
170
2845164447410820947
340
Jarek
0,20,24,44,60,66,80,84,86,90,104,110,126,146,150,170
0,146,44,150,110,84,66,80,126,20,170,24,104,90,60,86
3
16
172
2005475517332717239
344
Jarek
0,10,18,24,28,34,42,52,120,130,138,144,148,154,162,172
0,154,28,162,148,42,120,34,144,10,172,18,52,138,24,130
3
16
172
2416906123161566167
344
Jarek
0,10,36,46,60,66,70,76,96,102,106,112,126,136,162,172
0,136,46,162,112,96,66,70,126,10,172,36,106,102,60,76
3
16
172
3320148889158253969
344
Jarek
0,12,22,34,48,60,70,82,90,102,112,124,138,150,160,172
0,160,34,150,124,60,90,70,138,22,172,12,82,102,48,112
3
16
172
6943668427522772461
344
Jarek
0,10,36,46,60,66,70,76,96,102,106,112,126,136,162,172
0,136,46,162,112,96,66,70,126,10,172,36,106,102,60,76
3
16
172
822148161893494519
344
Jarek
0,12,28,30,40,42,58,70,102,114,130,132,142,144,160,172
0,160,40,144,142,42,102,58,132,28,172,12,70,114,30,130
3
16
176
1759943151645258947
352
Jarek
0,2,12,14,42,44,54,56,120,122,132,134,162,164,174,176
0,164,14,174,134,54,120,44,162,2,176,12,56,132,42,122
3
16
176
244890665002815107
352
Jarek
0,6,24,30,66,72,80,86,90,96,104,110,146,152,170,176
0,170,86,96,152,30,66,104,90,80,176,6,110,72,24,146
3
16
176
256734097976480057
352
Jarek
0,6,20,26,66,72,84,86,90,92,104,110,150,156,170,176
0,170,26,156,110,72,84,86,150,20,176,6,92,90,66,104
3
16
176
3954044488109204897
352
Jarek
0,6,14,20,30,36,44,50,126,132,140,146,156,162,170,176
0,170,20,162,146,36,126,44,156,14,176,6,50,132,30,140
3
16
176
5030319683017841117
352
Jarek
0,6,14,20,66,72,80,86,90,96,104,110,156,162,170,176
0,170,20,162,110,72,90,80,156,14,176,6,86,96,66,104
3
16
176
6926619233106731357
352
Jarek
0,6,20,26,66,72,84,86,90,92,104,110,150,156,170,176
0,170,26,156,110,72,84,86,150,20,176,6,92,90,66,104
3
16
178
125857555941397681
356
Jarek
0,10,42,48,52,58,78,88,90,100,120,126,130,136,168,178
0,136,52,168,130,90,78,58,126,10,178,42,100,120,48,88
3
16
178
1638289267064976151
356
Jarek
0,10,30,40,42,52,72,82,96,106,126,136,138,148,168,178
0,148,40,168,136,72,96,52,138,10,178,30,82,126,42,106
3
16
178
247107577536578131
356
Jarek
0,6,10,16,42,48,52,58,120,126,130,136,162,168,172,178
0,172,16,168,136,48,120,52,162,10,178,6,58,126,42,130
3
16
178
3580535146422008413
356
Jarek
0,4,30,34,60,64,84,88,90,94,114,118,144,148,174,178
0,148,34,174,118,90,84,64,144,4,178,30,94,114,60,88
3
16
178
4527086954095994641
356
Jarek
0,12,30,40,42,52,70,82,96,108,126,136,138,148,166,178
0,166,52,138,148,42,96,70,126,40,178,12,82,108,30,136
3
16
182
3300518070945984359
364
Jarek
0,18,20,38,60,78,80,84,98,102,104,122,144,162,164,182
0,164,38,162,122,78,84,80,144,20,182,18,98,102,60,104
3
16
182
6234691012623925289
364
Jarek
0,8,12,20,42,50,54,62,120,128,132,140,162,170,174,182
0,170,20,174,140,54,120,50,162,8,182,12,62,132,42,128
3
16
184
2104650358729544023
368
Jarek
0,6,34,40,60,66,84,90,94,100,118,124,144,150,178,184
0,178,40,150,124,66,84,94,144,34,184,6,100,90,60,118
3
16
184
2962098063724222219
368
Jarek
0,12,30,40,42,52,70,82,102,114,132,142,144,154,172,184
0,172,52,144,154,42,102,70,132,40,184,12,82,114,30,142
3
16
188
1709642327471063801
376
Jarek
0,2,30,32,60,62,90,92,96,98,126,128,156,158,186,188
0,158,32,186,128,90,96,62,156,2,188,30,92,126,60,98
3
16
188
3464369515329419603
376
Jarek
0,14,30,44,60,74,84,90,98,104,114,128,144,158,174,188
0,158,44,174,128,90,84,74,144,14,188,30,104,114,60,98
3
16
188
4308119447085349781
376
Jarek
0,6,30,36,42,48,72,78,110,116,140,146,152,158,182,188
0,182,116,78,158,36,42,140,72,110,188,6,146,48,30,152
3
16
188
4789518210750386921
376
Jarek
0,20,36,42,56,62,78,90,98,110,126,132,146,152,168,188
0,152,56,168,146,78,90,62,132,20,188,36,98,126,42,110
3
16
190
136574805190153411
380
Jarek
0,28,30,42,58,70,72,90,100,118,120,132,148,160,162,190
0,160,58,162,148,72,90,70,132,28,190,30,100,120,42,118
3
16
190
2970792733217448709
380
Jarek
0,10,30,40,42,52,72,82,108,118,138,148,150,160,180,190
0,160,40,180,148,72,108,52,150,10,190,30,82,138,42,118
3
16
190
3259520866813541389
380
Jarek
0,22,30,48,52,70,78,90,100,112,120,138,142,160,168,190
0,160,52,168,142,78,90,70,138,22,190,30,100,120,48,112
3
16
190
5378927763817198771
380
Jarek
0,10,42,52,60,70,78,88,102,112,120,130,138,148,180,190
0,148,52,180,130,102,78,70,138,10,190,42,112,120,60,88
3
16
190
543046371789268681
380
Jarek
0,10,42,52,60,70,78,88,102,112,120,130,138,148,180,190
0,148,52,180,130,102,78,70,138,10,190,42,112,120,60,88
3
16
190
6396263061532754749
380
Jarek
0,10,30,40,42,52,72,82,108,118,138,148,150,160,180,190
0,160,40,180,148,72,108,52,150,10,190,30,82,138,42,118
3
16
194
30870839193095093
388
Jarek
0,14,30,44,66,80,84,96,98,110,114,128,150,164,180,194
0,164,44,180,128,96,84,80,150,14,194,30,110,114,66,98
3
16
194
4051849896719838407
388
Jarek
0,12,20,32,42,54,62,74,120,132,140,152,162,174,182,194
0,182,32,174,152,54,120,62,162,20,194,12,74,132,42,140
3
16
194
52820951046679703
388
Jarek
0,20,24,44,66,84,86,90,104,108,110,128,150,170,174,194
0,170,44,174,128,90,84,86,150,20,194,24,110,108,66,104
3
16
194
7300094584443422507
388
Jarek
0,12,20,32,42,54,62,74,120,132,140,152,162,174,182,194
0,182,32,174,152,54,120,62,162,20,194,12,74,132,42,140
3
16
196
197328576729627247
392
Jarek
0,4,42,46,60,64,90,94,102,106,132,136,150,154,192,196
0,154,46,192,136,102,90,64,150,4,196,42,106,132,60,94
3
16
196
2120809189473200557
392
Jarek
0,24,36,60,66,70,90,94,102,106,126,130,136,160,172,196
0,172,94,126,160,60,66,106,102,70,196,24,130,90,36,136
3
16
196
2609488448804959987
392
Jarek
0,12,30,40,42,52,70,82,114,126,144,154,156,166,184,196
0,184,52,156,166,42,114,70,144,40,196,12,82,126,30,154
3
16
196
4418808046547962567
392
Jarek
0,12,22,34,42,54,64,76,120,132,142,154,162,174,184,196
0,184,34,174,154,54,120,64,162,22,196,12,76,132,42,142
3
16
196
700770556458849277
392
Jarek
0,10,24,34,42,52,66,76,120,130,144,154,162,172,186,196
0,172,34,186,154,66,120,52,162,10,196,24,76,144,42,130
3
16
200
4930532549661487103
400
Jarek
0,14,30,44,60,74,90,96,104,110,126,140,156,170,186,200
0,170,44,186,140,90,96,74,156,14,200,30,104,126,60,110
3
16
200
4960536274955735003
400
Jarek
0,14,36,50,60,74,90,96,104,110,126,140,150,164,186,200
0,164,50,186,140,96,90,74,150,14,200,36,110,126,60,104
3
16
202
1161886340216531929
404
Jarek
0,18,42,52,60,70,90,94,108,112,132,142,150,160,184,202
0,184,70,150,160,60,90,94,132,52,202,18,112,108,42,142
3
16
202
1260702715810312429
404
Jarek
0,28,30,42,58,70,72,100,102,130,132,144,160,172,174,202
0,172,58,174,160,72,102,70,144,28,202,30,100,132,42,130
3
16
202
3526298299878274429
404
Jarek
0,12,42,54,70,78,82,90,112,120,124,132,148,160,190,202
0,190,82,132,160,54,78,112,120,70,202,12,124,90,42,148
3
16
206
1184206362130402157
412
Jarek
0,30,54,56,66,84,86,96,110,120,122,140,150,152,176,206
0,176,86,150,152,84,66,110,120,56,206,30,140,96,54,122
3
16
206
1274498753259676961
412
Jarek
0,18,20,38,60,78,80,98,108,126,128,146,168,186,188,206
0,188,38,186,146,78,108,80,168,20,206,18,98,126,60,128
3
16
206
1349849386699416773
412
Jarek
0,18,48,60,66,78,80,98,108,126,128,140,146,158,188,206
0,188,98,126,158,66,60,128,108,80,206,18,146,78,48,140
3
16
206
1944662984243678687
412
Jarek
0,32,42,60,72,74,92,102,104,114,132,134,146,164,174,206
0,164,74,174,146,102,72,92,132,32,206,42,134,114,60,104
3
16
206
2454945853671281441
412
Jarek
0,18,20,38,60,78,80,98,108,126,128,146,168,186,188,206
0,188,38,186,146,78,108,80,168,20,206,18,98,126,60,128
3
16
206
4593033835811242937
412
Jarek
0,26,30,56,66,84,92,96,110,114,122,140,150,176,180,206
0,176,56,180,140,96,84,92,150,26,206,30,122,114,66,110
3
16
206
5512467165717387017
412
Jarek
0,2,30,32,42,44,72,74,132,134,162,164,174,176,204,206
0,176,32,204,164,72,132,44,174,2,206,30,74,162,42,134
3
16
206
584975972044768607
412
Jarek
0,6,50,56,66,72,84,90,116,122,134,140,150,156,200,206
0,200,56,156,140,72,84,116,150,50,206,6,122,90,66,134
3
16
206
6143618019866182577
412
Jarek
0,12,24,30,36,42,54,66,140,152,164,170,176,182,194,206
0,194,152,66,182,36,30,164,54,140,206,12,176,42,24,170
3
16
208
2657619911321552311
416
Jarek
0,10,36,46,60,70,96,102,106,112,138,148,162,172,198,208
0,172,46,198,148,96,102,70,162,10,208,36,106,138,60,112
3
16
208
3240195864372286321
416
Jarek
0,36,40,42,76,78,82,90,118,126,130,132,166,168,172,208
0,172,76,168,166,78,90,82,132,40,208,36,118,126,42,130
3
16
208
3288578117539863121
416
Jarek
0,28,30,42,58,70,72,100,108,136,138,150,166,178,180,208
0,178,58,180,166,72,108,70,150,28,208,30,100,138,42,136
3
16
208
4648589081277508711
416
Jarek
0,10,18,28,60,70,78,88,120,130,138,148,180,190,198,208
0,190,28,198,148,78,120,70,180,10,208,18,88,138,60,130
3
16
208
5143686234236121601
416
Jarek
0,18,22,40,60,78,82,100,108,126,130,148,168,186,190,208
0,190,40,186,148,78,108,82,168,22,208,18,100,126,60,130
3
16
208
5924013319890276241
416
Jarek
0,12,30,42,66,78,96,100,108,112,130,142,166,178,196,208
0,196,112,108,178,42,66,130,96,100,208,12,142,78,30,166
3
16
208
6460757831849769991
416
Jarek
0,12,48,60,70,78,82,90,118,126,130,138,148,160,196,208
0,196,82,138,160,60,78,118,126,70,208,12,130,90,48,148
3
16
208
7129286091391865041
416
Jarek
0,28,42,60,70,78,88,102,106,120,130,138,148,166,180,208
0,166,70,180,148,102,78,88,138,28,208,42,130,120,60,106
3
16
208
7302012435009585511
416
Jarek
0,12,60,66,70,72,78,82,126,130,136,138,142,148,196,208
0,196,82,138,148,72,66,130,126,70,208,12,142,78,60,136
3
16
212
1025519173619653079
424
Jarek
0,2,42,44,78,80,90,92,120,122,132,134,168,170,210,212
0,170,44,210,134,120,90,80,168,2,212,42,122,132,78,92
3
16
212
119890755200639999
424
Jarek
0,2,42,44,78,80,90,92,120,122,132,134,168,170,210,212
0,170,44,210,134,120,90,80,168,2,212,42,122,132,78,92
3
16
212
2998166279211456467
424
Jarek
0,6,30,36,50,56,80,86,126,132,156,162,176,182,206,212
0,206,56,162,182,36,126,80,156,50,212,6,86,132,30,176
3
16
212
3348119580392456939
424
Jarek
0,14,24,38,84,90,98,104,108,114,122,128,174,188,198,212
0,188,38,198,128,108,90,98,174,14,212,24,122,114,84,104
3
16
212
483563057397037421
424
Jarek
0,12,30,42,60,72,90,102,110,122,140,152,170,182,200,212
0,200,122,102,182,42,60,140,90,110,212,12,152,72,30,170
3
16
212
5768551885430844239
424
Jarek
0,14,30,44,78,90,92,104,108,120,122,134,168,182,198,212
0,182,44,198,134,108,90,92,168,14,212,30,122,120,78,104
3
16
214
121677799699349479
428
Jarek
0,10,24,34,60,70,84,94,120,130,144,154,180,190,204,214
0,190,34,204,154,84,120,70,180,10,214,24,94,144,60,130
3
16
214
6145034310642442003
428
Jarek
0,10,60,66,70,76,78,88,126,136,138,144,148,154,204,214
0,154,70,204,148,126,78,76,144,10,214,60,136,138,66,88
3
16
214
6473064433226665033
428
Jarek
0,18,30,48,70,88,96,100,114,118,126,144,166,184,196,214
0,196,88,144,184,48,96,100,126,70,214,18,118,114,30,166
3
16
214
6860133665885637793
428
Jarek
0,10,18,28,66,76,84,94,120,130,138,148,186,196,204,214
0,196,28,204,148,84,120,76,186,10,214,18,94,138,66,130
3
16
214
826028861488453243
428
Jarek
0,36,48,60,70,84,96,106,108,118,130,144,154,166,178,214
0,178,106,144,166,84,60,118,108,70,214,36,154,96,48,130
3
16
218
1792052806371623099
436
Jarek
0,20,30,50,78,90,98,108,110,120,128,140,168,188,198,218
0,188,50,198,140,108,90,98,168,20,218,30,128,120,78,110
3
16
218
3699486792695636141
436
Jarek
0,6,30,36,50,56,80,86,132,138,162,168,182,188,212,218
0,212,56,168,188,36,132,80,162,50,218,6,86,138,30,182
3
16
218
6837944486582519201
436
Jarek
0,12,30,36,42,48,66,78,140,152,170,176,182,188,206,218
0,206,152,78,188,42,36,170,66,140,218,12,182,48,30,176
3
16
218
6837953580968433689
436
Jarek
0,20,30,50,78,90,98,108,110,120,128,140,168,188,198,218
0,188,50,198,140,108,90,98,168,20,218,30,128,120,78,110
3
16
218
6903481620265522931
436
Jarek
0,6,30,36,56,62,86,92,126,132,156,162,182,188,212,218
0,212,62,162,188,36,126,86,156,56,218,6,92,132,30,182
3
16
220
1104766616538404359
440
Jarek
0,30,40,42,70,72,82,108,112,138,148,150,178,180,190,220
0,190,70,180,178,72,108,82,150,40,220,30,112,138,42,148
3
16
220
1977443805899164879
440
Jarek
0,30,40,42,70,72,82,108,112,138,148,150,178,180,190,220
0,190,70,180,178,72,108,82,150,40,220,30,112,138,42,148
3
16
220
3516575775430342843
440
Jarek
0,30,40,66,70,84,96,106,114,124,136,150,154,180,190,220
0,190,70,180,154,96,84,106,150,40,220,30,136,114,66,124
3
16
220
6144860086481473681
440
Jarek
0,10,42,48,52,58,90,100,120,130,162,168,172,178,210,220
0,178,52,210,172,90,120,58,168,10,220,42,100,162,48,130
3
16
224
1353204408659434493
448
Jarek
0,24,30,54,84,86,108,110,114,116,138,140,170,194,200,224
0,200,110,138,194,54,84,116,114,86,224,24,140,108,30,170
3
16
224
1365986292501549617
448
Jarek
0,30,42,62,72,90,92,104,120,132,134,152,162,182,194,224
0,194,92,162,182,72,90,104,132,62,224,30,134,120,42,152
3
16
224
1836717961457293283
448
Jarek
0,24,50,66,74,84,90,108,116,134,140,150,158,174,200,224
0,200,74,174,158,90,84,116,150,50,224,24,140,108,66,134
3
16
224
3211471911994853333
448
Jarek
0,26,30,54,56,80,84,110,114,140,144,168,170,194,198,224
0,194,56,198,170,84,114,80,168,26,224,30,110,144,54,140
3
16
224
3535827318958835063
448
Jarek
0,18,20,38,60,78,80,98,126,144,146,164,186,204,206,224
0,206,38,204,164,78,126,80,186,20,224,18,98,144,60,146
3
16
224
3676207202733217613
448
Jarek
0,24,30,54,84,86,108,110,114,116,138,140,170,194,200,224
0,200,110,138,194,54,84,116,114,86,224,24,140,108,30,170
3
16
224
3925937432649879263
448
Jarek
0,14,30,44,66,80,96,110,114,128,144,158,180,194,210,224
0,194,44,210,158,96,114,80,180,14,224,30,110,144,66,128
3
16
224
4072432651991186423
448
Jarek
0,6,14,20,30,36,44,50,174,180,188,194,204,210,218,224
0,218,20,210,194,36,174,44,204,14,224,6,50,180,30,188
3
16
224
858132626286456923
448
Jarek
0,6,50,56,78,84,90,96,128,134,140,146,168,174,218,224
0,218,56,174,146,84,90,128,168,50,224,6,134,96,78,140
3
16
226
1348850943815472523
452
Jarek
0,30,40,60,70,90,96,100,126,130,136,156,166,186,196,226
0,196,70,186,166,90,96,100,156,40,226,30,130,126,60,136
3
16
226
2743915961319802891
452
Jarek
0,18,30,40,48,58,70,88,138,156,168,178,186,196,208,226
0,208,58,186,196,48,138,70,168,40,226,18,88,156,30,178
3
16
226
4386788379691729507
452
Jarek
0,12,30,42,70,82,100,112,114,126,144,156,184,196,214,226
0,214,82,156,196,42,114,100,144,70,226,12,112,126,30,184
3
16
226
5419298383390453957
452
Jarek
0,30,42,64,72,90,94,106,120,132,136,154,162,184,196,226
0,196,94,162,184,72,90,106,132,64,226,30,136,120,42,154
3
16
226
5428670927917516177
452
Jarek
0,6,16,22,84,90,100,106,120,126,136,142,204,210,220,226
0,220,22,210,142,90,120,100,204,16,226,6,106,126,84,136
3
16
226
6205884072061035403
452
Jarek
0,18,48,66,70,88,90,108,118,136,138,156,160,178,208,226
0,208,88,156,178,66,90,118,138,70,226,18,136,108,48,160
3
16
226
75219020808006217
452
Jarek
0,6,36,42,70,76,106,112,114,120,150,156,184,190,220,226
0,220,76,156,190,42,114,106,150,70,226,6,112,120,36,184
3
16
226
983481025217213137
452
Jarek
0,30,40,70,72,84,102,112,114,124,142,154,156,186,196,226
0,196,70,186,154,102,84,112,156,40,226,30,142,114,72,124
3
16
230
1013817764091131963
460
Jarek
0,20,36,56,84,90,104,110,120,126,140,146,174,194,210,230
0,194,56,210,146,120,90,104,174,20,230,36,140,126,84,110
3
16
230
1161621187363787303
460
Jarek
0,14,30,44,60,74,90,104,126,140,156,170,186,200,216,230
0,200,44,216,170,90,126,74,186,14,230,30,104,156,60,140
3
16
230
2746765265140004819
460
Jarek
0,8,12,20,90,98,102,110,120,128,132,140,210,218,222,230
0,218,20,222,140,102,120,98,210,8,230,12,110,132,90,128
3
16
230
2849547638609642513
460
Jarek
0,26,30,56,60,86,90,114,116,140,144,170,174,200,204,230
0,200,56,204,170,90,114,86,174,26,230,30,116,144,60,140
3
16
230
3284360538807735461
460
Jarek
0,12,20,32,78,90,98,110,120,132,140,152,198,210,218,230
0,218,32,210,152,90,120,98,198,20,230,12,110,132,78,140
3
16
230
3808061696393625101
460
Jarek
0,2,30,32,60,62,90,92,138,140,168,170,198,200,228,230
0,200,32,228,170,90,138,62,198,2,230,30,92,168,60,140
3
16
230
5554064482809089477
460
Jarek
0,14,60,66,74,80,90,104,126,140,150,156,164,170,216,230
0,170,74,216,164,126,90,80,156,14,230,60,140,150,66,104
3
16
230
6044277931660903169
460
Jarek
0,20,30,48,50,68,78,98,132,152,162,180,182,200,210,230
0,200,50,210,182,78,132,68,180,20,230,30,98,162,48,152
3
16
232
1949587307274796249
464
Jarek
0,22,42,64,78,90,100,112,120,132,142,154,168,190,210,232
0,190,64,210,154,120,90,100,168,22,232,42,142,132,78,112
3
16
232
2739157114418172739
464
Jarek
0,30,48,70,78,84,100,114,118,132,148,154,162,184,202,232
0,202,100,162,184,78,84,118,132,70,232,30,148,114,48,154
3
16
232
4669370734348914049
464
Jarek
0,10,42,52,60,70,102,112,120,130,162,172,180,190,222,232
0,190,52,222,172,102,120,70,180,10,232,42,112,162,60,130
3
16
232
4769304739639109671
464
Jarek
0,36,60,66,70,96,102,106,126,130,136,162,166,172,196,232
0,196,106,162,172,96,66,130,126,70,232,36,166,102,60,136
3
16
232
4854227454119376679
464
Jarek
0,10,24,34,90,100,108,114,118,124,132,142,198,208,222,232
0,208,34,222,142,114,108,100,198,10,232,24,124,132,90,118
3
16
232
5430518283273066469
464
Jarek
0,22,30,48,52,70,78,100,132,154,162,180,184,202,210,232
0,202,52,210,184,78,132,70,180,22,232,30,100,162,48,154
3
16
232
6455528034823901701
464
Jarek
0,6,66,70,72,76,90,96,136,142,156,160,162,166,226,232
0,226,76,162,166,72,90,136,156,70,232,6,142,96,66,160
3
16
232
942188212207026139
464
Jarek
0,18,24,42,60,78,84,102,130,148,154,172,190,208,214,232
0,214,148,102,208,42,60,154,84,130,232,18,172,78,24,190
3
16
236
1748611243534828247
472
Jarek
0,24,26,50,60,84,86,110,126,150,152,176,186,210,212,236
0,212,50,210,176,84,126,86,186,26,236,24,110,150,60,152
3
16
236
4814184375353087117
472
Jarek
0,12,42,50,54,62,92,104,132,144,174,182,186,194,224,236
0,224,62,186,194,54,132,92,174,50,236,12,104,144,42,182
3
16
236
5172657399101044427
472
Jarek
0,42,50,60,84,92,102,110,126,134,144,152,176,186,194,236
0,194,92,186,176,102,84,110,144,50,236,42,152,126,60,134
3
16
236
597511709585678627
472
Jarek
0,6,20,26,36,42,56,62,174,180,194,200,210,216,230,236
0,230,26,216,200,42,174,56,210,20,236,6,62,180,36,194
3
16
238
1111443560879387521
476
Jarek
0,42,60,66,70,102,108,112,126,130,136,168,172,178,196,238
0,196,112,168,178,102,66,130,126,70,238,42,172,108,60,136
3
16
238
1684157406173732383
476
Jarek
0,4,24,28,60,64,84,88,150,154,174,178,210,214,234,238
0,214,28,234,178,84,150,64,210,4,238,24,88,174,60,154
3
16
238
1722099695477586181
476
Jarek
0,12,28,30,40,42,58,70,168,180,196,198,208,210,226,238
0,226,40,210,208,42,168,58,198,28,238,12,70,180,30,196
3
16
238
2870027314708591981
476
Jarek
0,30,42,58,72,88,100,108,130,138,150,166,180,196,208,238
0,208,88,180,196,72,108,100,150,58,238,30,130,138,42,166
3
16
238
3330362879526931471
476
Jarek
0,18,42,60,70,88,108,112,126,130,150,168,178,196,220,238
0,220,88,168,196,60,108,112,150,70,238,18,130,126,42,178
3
16
238
520330171849862431
476
Jarek
0,12,18,28,30,40,46,58,180,192,198,208,210,220,226,238
0,226,40,210,220,30,180,46,198,28,238,12,58,192,18,208
3
16
238
5990853535865174191
476
Jarek
0,10,42,52,78,88,108,118,120,130,150,160,186,196,228,238
0,196,52,228,160,120,108,88,186,10,238,42,130,150,78,118
3
16
238
6245958363138445849
476
Jarek
0,10,24,34,84,94,108,118,120,130,144,154,204,214,228,238
0,214,34,228,154,108,120,94,204,10,238,24,118,144,84,130
3
16
238
6828326467141699951
476
Jarek
0,12,36,48,70,82,106,118,120,132,156,168,190,202,226,238
0,226,82,168,202,48,120,106,156,70,238,12,118,132,36,190
3
16
242
470383969303868849
484
Jarek
0,12,48,50,60,62,98,110,132,144,180,182,192,194,230,242
0,230,62,192,194,60,132,98,180,50,242,12,110,144,48,182
3
16
242
505751676098073269
484
Jarek
0,12,30,42,60,72,90,102,140,152,170,182,200,212,230,242
0,230,152,102,212,42,60,170,90,140,242,12,182,72,30,200
3
16
242
6276472529120900609
484
Jarek
0,18,42,60,72,90,110,114,128,132,152,170,182,200,224,242
0,224,128,132,200,60,72,152,114,110,242,18,170,90,42,182
3
16
242
854802325142674739
484
Jarek
0,12,42,54,68,80,110,120,122,132,162,174,188,200,230,242
0,230,80,174,200,54,120,110,162,68,242,12,122,132,42,188
3
16
244
1422162363915832963
488
Jarek
0,36,40,48,76,84,88,120,124,156,160,168,196,204,208,244
0,208,76,204,196,84,120,88,168,40,244,36,124,156,48,160
3
16
244
1695576612355403647
488
Jarek
0,24,60,70,84,90,94,114,130,150,154,160,174,184,220,244
0,220,94,174,184,84,90,130,150,70,244,24,154,114,60,160
3
16
244
193747830997541713
488
Jarek
0,18,28,46,60,78,88,106,138,156,166,184,198,216,226,244
0,226,46,216,184,78,138,88,198,28,244,18,106,156,60,166
3
16
244
2877019027302762913
488
Jarek
0,6,54,60,70,76,114,120,124,130,168,174,184,190,238,244
0,238,76,174,190,60,114,124,168,70,244,6,130,120,54,184
3
16
244
2996774562519515923
488
Jarek
0,40,54,66,84,94,106,120,124,138,150,160,178,190,204,244
0,190,94,204,178,120,84,106,150,40,244,54,160,138,66,124
3
16
244
4590873012397692739
488
Jarek
0,12,42,54,70,82,112,120,124,132,162,174,190,202,232,244
0,232,82,174,202,54,120,112,162,70,244,12,124,132,42,190
3
16
244
5229591821941140757
488
Jarek
0,12,42,54,70,82,112,120,124,132,162,174,190,202,232,244
0,232,82,174,202,54,120,112,162,70,244,12,124,132,42,190
3
16
244
593606097226087453
488
Jarek
0,18,40,58,60,78,100,118,126,144,166,184,186,204,226,244
0,226,58,204,184,78,126,100,186,40,244,18,118,144,60,166
3
16
244
5986294670848092703
488
Jarek
0,6,40,46,84,90,114,120,124,130,154,160,198,204,238,244
0,238,46,204,160,90,114,124,198,40,244,6,130,120,84,154
3
16
244
6095019077256106423
488
Jarek
0,18,66,70,84,88,90,108,136,154,156,160,174,178,226,244
0,226,88,174,178,84,90,136,156,70,244,18,154,108,66,160
3
16
244
6218513221849930999
488
Jarek
0,30,40,60,70,90,100,114,130,144,154,174,184,204,214,244
0,214,70,204,184,90,114,100,174,40,244,30,130,144,60,154
3
16
248
2187508254185189591
496
Jarek
0,42,60,66,80,102,108,122,126,140,146,168,182,188,206,248
0,206,122,168,188,102,66,140,126,80,248,42,182,108,60,146
3
16
250
4372961435616492067
500
Jarek
0,6,24,30,70,76,94,100,150,156,174,180,220,226,244,250
0,244,76,180,226,30,150,94,174,70,250,6,100,156,24,220
3
16
250
6632564174879375731
500
Jarek
0,18,60,70,78,88,102,120,130,148,162,172,180,190,232,250
0,232,88,180,190,78,102,130,162,70,250,18,148,120,60,172
3
16
250
6895412835092216299
500
Jarek
0,12,28,40,90,102,118,120,130,132,148,160,210,222,238,250
0,238,40,222,160,102,120,118,210,28,250,12,130,132,90,148
3
16
254
1548843198482789639
508
Jarek
0,30,42,50,72,80,92,122,132,162,174,182,204,212,224,254
0,224,80,204,212,72,132,92,174,50,254,30,122,162,42,182
3
16
254
4571312291409422243
508
Jarek
0,20,24,44,60,80,84,104,150,170,174,194,210,230,234,254
0,230,44,234,194,84,150,80,210,20,254,24,104,174,60,170
3
16
256
183654607712944657
512
Jarek
0,24,42,60,66,84,102,126,130,154,172,190,196,214,232,256
0,232,154,126,214,66,60,172,102,130,256,24,196,84,42,190
3
16
256
1991122514241536437
512
Jarek
0,12,42,54,100,102,112,114,142,144,154,156,202,214,244,256
0,244,112,156,214,54,102,142,144,100,256,12,154,114,42,202
3
16
256
2370832941679372567
512
Jarek
0,12,22,34,90,102,112,124,132,144,154,166,222,234,244,256
0,244,34,234,166,102,132,112,222,22,256,12,124,144,90,154
3
16
256
2901380697433271107
512
Jarek
0,40,42,54,82,94,96,120,136,160,162,174,202,214,216,256
0,214,82,216,202,96,120,94,174,40,256,42,136,162,54,160
3
16
256
3754487283109755883
512
Jarek
0,18,48,66,70,88,118,120,136,138,168,186,190,208,238,256
0,238,88,186,208,66,120,118,168,70,256,18,136,138,48,190
3
16
256
5386876445300866597
512
Jarek
0,30,42,72,82,102,112,124,132,144,154,174,184,214,226,256
0,226,112,174,214,72,102,124,144,82,256,30,154,132,42,184
3
16
256
5502527692824783637
512
Jarek
0,24,46,60,70,84,106,126,130,150,172,186,196,210,232,256
0,232,70,210,196,84,126,106,186,46,256,24,130,150,60,172
3
16
256
724964620942114237
512
Jarek
0,12,42,54,90,102,112,124,132,144,154,166,202,214,244,256
0,244,124,144,214,54,90,154,132,112,256,12,166,102,42,202
3
16
260
1115116970440210853
520
Jarek
0,20,66,84,86,90,104,110,150,156,170,174,176,194,240,260
0,194,86,240,176,150,90,104,174,20,260,66,170,156,84,110
3
16
260
172234415276027219
520
Jarek
0,20,42,48,62,68,90,110,150,170,192,198,212,218,240,260
0,218,62,240,212,90,150,68,198,20,260,42,110,192,48,170
3
16
260
4136769267384711527
520
Jarek
0,26,30,56,84,110,114,120,140,146,150,176,204,230,234,260
0,230,56,234,176,114,120,110,204,26,260,30,140,150,84,146
3
16
260
4949096256248126783
520
Jarek
0,30,36,66,84,110,114,120,140,146,150,176,194,224,230,260
0,230,140,150,224,66,84,146,120,110,260,30,176,114,36,194
3
16
260
576195018029325059
520
Jarek
0,18,60,78,80,98,102,120,140,158,162,180,182,200,242,260
0,242,98,180,200,78,102,140,162,80,260,18,158,120,60,182
3
16
260
970723521259217267
520
Jarek
0,20,24,44,66,86,90,110,150,170,174,194,216,236,240,260
0,236,44,240,194,90,150,86,216,20,260,24,110,174,66,170
3
16
262
1394323695407632369
524
Jarek
0,22,30,52,78,100,108,130,132,154,162,184,210,232,240,262
0,232,52,240,184,108,132,100,210,22,262,30,130,162,78,154
3
16
262
523223163845273719
524
Jarek
0,18,42,60,70,88,112,130,132,150,174,192,202,220,244,262
0,244,88,192,220,60,132,112,174,70,262,18,130,150,42,202
3
16
262
6739882649149671649
524
Jarek
0,18,42,60,90,108,112,130,132,150,154,172,202,220,244,262
0,244,130,150,220,60,90,154,132,112,262,18,172,108,42,202
3
16
262
90658333138283839
524
Jarek
0,40,42,48,82,88,90,130,132,172,174,180,214,220,222,262
0,220,82,222,214,90,132,88,180,40,262,42,130,174,48,172
3
16
266
1662966605198077337
532
Jarek
0,14,42,56,90,104,120,132,134,146,162,176,210,224,252,266
0,224,56,252,176,132,120,104,210,14,266,42,146,162,90,134
3
16
266
3674413084938709037
532
Jarek
0,12,30,42,84,96,114,126,140,152,170,182,224,236,254,266
0,254,152,126,236,42,84,170,114,140,266,12,182,96,30,224
3
16
266
4276861062256425977
532
Jarek
0,24,36,60,66,90,102,126,140,164,176,200,206,230,242,266
0,242,164,126,230,60,66,176,102,140,266,24,200,90,36,206
3
16
266
5388267171062582297
532
Jarek
0,24,30,54,72,96,102,126,140,164,170,194,212,236,242,266
0,242,164,126,236,54,72,170,102,140,266,24,194,96,30,212
3
16
266
65659672001596307
532
Jarek
0,24,60,66,84,90,116,126,140,150,176,182,200,206,242,266
0,242,140,150,206,84,66,176,126,116,266,24,200,90,60,182
3
16
266
6894646511664746207
532
Jarek
0,24,36,60,66,90,102,126,140,164,176,200,206,230,242,266
0,242,164,126,230,60,66,176,102,140,266,24,200,90,36,206
3
16
268
331797868801711441
536
Jarek
0,22,30,52,90,112,120,126,142,148,156,178,216,238,246,268
0,238,52,246,178,120,126,112,216,22,268,30,142,156,90,148
3
16
268
3551657267160637651
536
Jarek
0,30,48,70,78,100,118,120,148,150,168,190,198,220,238,268
0,238,100,198,220,78,120,118,168,70,268,30,148,150,48,190
3
16
268
3561150466636039561
536
Jarek
0,36,42,60,78,96,102,130,138,166,172,190,208,226,232,268
0,232,166,138,226,78,60,172,102,130,268,36,208,96,42,190
3
16
268
392952504670299499
536
Jarek
0,30,48,70,78,100,118,120,148,150,168,190,198,220,238,268
0,238,100,198,220,78,120,118,168,70,268,30,148,150,48,190
3
16
268
573863571825332491
536
Jarek
0,30,42,72,76,106,118,120,148,150,162,192,196,226,238,268
0,238,106,192,226,72,120,118,162,76,268,30,148,150,42,196
3
16
268
7238777982941181361
536
Jarek
0,60,66,70,72,126,130,132,136,138,142,196,198,202,208,268
0,208,130,198,202,126,72,136,138,70,268,60,196,132,66,142
3
16
272
2055009868507385459
544
Jarek
0,42,48,50,90,92,98,132,140,174,180,182,222,224,230,272
0,230,92,222,224,90,132,98,180,50,272,42,140,174,48,182
3
16
272
213896077666130459
544
Jarek
0,20,42,62,90,110,120,132,140,152,162,182,210,230,252,272
0,230,62,252,182,132,120,110,210,20,272,42,152,162,90,140
3
16
272
3617374549635817109
544
Jarek
0,38,42,80,90,102,128,132,140,144,170,182,192,230,234,272
0,230,80,234,182,132,102,128,192,38,272,42,170,144,90,140
3
16
272
5051144543625838907
544
Jarek
0,6,30,36,90,96,120,126,146,152,176,182,236,242,266,272
0,266,152,126,242,36,90,176,120,146,272,6,182,96,30,236
3
16
274
1161374752701595303
548
Jarek
0,18,60,70,78,88,126,130,144,148,186,196,204,214,256,274
0,256,88,204,214,78,126,130,186,70,274,18,148,144,60,196
3
16
274
146740177075293973
548
Jarek
0,30,40,66,70,96,106,136,138,168,178,204,208,234,244,274
0,244,70,234,208,96,138,106,204,40,274,30,136,168,66,178
3
16
274
3069372153044336233
548
Jarek
0,34,66,84,90,100,118,124,150,156,174,184,190,208,240,274
0,208,100,240,190,150,90,118,174,34,274,66,184,156,84,124
3
16
274
3326474066397256183
548
Jarek
0,30,36,40,66,70,76,106,168,198,204,208,234,238,244,274
0,244,70,234,238,66,168,76,204,40,274,30,106,198,36,208
3
16
274
5075754267051808903
548
Jarek
0,6,30,36,84,90,114,120,154,160,184,190,238,244,268,274
0,268,160,120,244,36,84,184,114,154,274,6,190,90,30,238
3
16
274
7002957165963385603
548
Jarek
0,4,60,64,84,88,126,130,144,148,186,190,210,214,270,274
0,214,64,270,190,144,126,88,210,4,274,60,148,186,84,130
3
16
278
2202676356250302461
556
Jarek
0,6,36,42,96,102,132,138,140,146,176,182,236,242,272,278
0,272,146,138,242,42,96,176,132,140,278,6,182,102,36,236
3
16
278
2657370364046282393
556
Jarek
0,14,54,68,90,104,120,134,144,158,174,188,210,224,264,278
0,224,68,264,188,144,120,104,210,14,278,54,158,174,90,134
3
16
278
4303654267764620861
556
Jarek
0,30,36,66,72,102,108,138,140,170,176,206,212,242,248,278
0,248,170,138,242,66,72,176,108,140,278,30,206,102,36,212
3
16
278
4751478919613173733
556
Jarek
0,20,24,44,84,104,108,128,150,170,174,194,234,254,258,278
0,254,44,258,194,108,150,104,234,20,278,24,128,174,84,170
3
16
280
108978576749718703
560
Jarek
0,30,66,70,96,100,114,136,144,166,180,184,210,214,250,280
0,250,100,210,214,96,114,136,180,70,280,30,166,144,66,184
3
16
280
1152127746876591217
560
Jarek
0,10,36,46,84,94,120,130,150,160,186,196,234,244,270,280
0,244,46,270,196,120,150,94,234,10,280,36,130,186,84,160
3
16
280
2381171917182771109
560
Jarek
0,42,60,88,90,102,130,132,148,150,178,190,192,220,238,280
0,238,130,192,220,102,90,148,150,88,280,42,190,132,60,178
3
16
280
3000609520074222541
560
Jarek
0,18,60,78,90,108,112,130,150,168,172,190,202,220,262,280
0,262,130,168,220,78,90,172,150,112,280,18,190,108,60,202
3
16
280
3919037389119084523
560
Jarek
0,30,40,70,84,114,124,126,154,156,166,196,210,240,250,280
0,250,70,240,196,114,126,124,210,40,280,30,154,156,84,166
3
16
280
6232062498622812853
560
Jarek
0,16,30,46,84,100,114,130,150,166,180,196,234,250,264,280
0,250,46,264,196,114,150,100,234,16,280,30,130,180,84,166
3
16
280
632878876266807217
560
Jarek
0,4,60,64,90,94,126,130,150,154,186,190,216,220,276,280
0,220,64,276,190,150,126,94,216,4,280,60,154,186,90,130
3
16
280
7288094093539270027
560
Jarek
0,46,60,84,90,106,130,136,144,150,174,190,196,220,234,280
0,220,106,234,196,144,90,130,174,46,280,60,190,150,84,136
3
16
280
87883992459664963
560
Jarek
0,46,60,84,90,106,130,136,144,150,174,190,196,220,234,280
0,220,106,234,196,144,90,130,174,46,280,60,190,150,84,136
3
16
282
1023267524341869299
564
Jarek
0,30,48,78,84,114,120,132,150,162,168,198,204,234,252,282
0,234,78,252,198,132,120,114,204,30,282,48,162,168,84,150
3
16
282
4367495951483424229
564
Jarek
0,12,18,30,42,54,60,72,210,222,228,240,252,264,270,282
0,264,30,270,240,60,210,54,252,12,282,18,72,228,42,222
3
16
284
135636084688856267
568
Jarek
0,42,50,60,92,102,110,132,152,174,182,192,224,234,242,284
0,242,92,234,224,102,132,110,192,50,284,42,152,174,60,182
3
16
284
2435735862630081383
568
Jarek
0,30,36,50,66,80,86,116,168,198,204,218,234,248,254,284
0,254,80,234,248,66,168,86,204,50,284,30,116,198,36,218
3
16
284
2466954089329721129
568
Jarek
0,12,42,54,90,102,132,140,144,152,182,194,230,242,272,284
0,272,152,144,242,54,90,182,132,140,284,12,194,102,42,230
3
16
284
3684541435048485383
568
Jarek
0,30,36,50,66,80,86,116,168,198,204,218,234,248,254,284
0,254,80,234,248,66,168,86,204,50,284,30,116,198,36,218
3
16
284
4276130160796247363
568
Jarek
0,30,56,78,86,108,120,134,150,164,176,198,206,228,254,284
0,254,86,228,206,108,120,134,198,56,284,30,164,150,78,176
3
16
284
501455933430730433
568
Jarek
0,18,56,60,74,78,116,134,150,168,206,210,224,228,266,284
0,266,74,228,224,78,150,116,210,56,284,18,134,168,60,206
3
16
284
5309126148402591029
568
Jarek
0,12,42,54,60,72,102,114,170,182,212,224,230,242,272,284
0,272,182,114,242,54,60,212,102,170,284,12,224,72,42,230
3
16
286
2529635432547452107
572
Jarek
0,40,42,82,90,114,130,132,154,156,172,196,204,244,246,286
0,244,82,246,196,132,114,130,204,40,286,42,172,156,90,154
3
16
286
2959720113369490771
572
Jarek
0,28,48,60,76,88,108,136,150,178,198,210,226,238,258,286
0,238,76,258,226,108,150,88,210,28,286,48,136,198,60,178
3
16
286
369819834773439997
572
Jarek
0,46,66,84,90,112,130,136,150,156,174,196,202,220,240,286
0,220,112,240,202,150,90,130,174,46,286,66,196,156,84,136
3
16
286
3901689256438869307
572
Jarek
0,24,46,66,70,90,112,136,150,174,196,216,220,240,262,286
0,262,70,240,220,90,150,112,216,46,286,24,136,174,66,196
3
16
286
605112013859711227
572
Jarek
0,30,42,72,84,114,126,130,156,160,172,202,214,244,256,286
0,256,160,156,244,72,84,172,126,130,286,30,202,114,42,214
3
16
286
64357903328718397
572
Jarek
0,6,10,16,126,132,136,142,144,150,154,160,270,276,280,286
0,280,16,276,160,132,144,136,270,10,286,6,142,150,126,154
3
16
286
973289520281889157
572
Jarek
0,42,54,70,96,112,120,124,162,166,174,190,216,232,244,286
0,244,112,216,232,96,120,124,174,70,286,42,166,162,54,190
3
16
290
2779503923057699879
580
Jarek
0,18,42,60,90,108,132,140,150,158,182,200,230,248,272,290
0,272,158,150,248,60,90,182,132,140,290,18,200,108,42,230
3
16
290
4293822215103776063
580
Jarek
0,30,36,66,90,120,126,134,156,164,170,200,224,254,260,290
0,260,164,156,254,66,90,170,126,134,290,30,200,120,36,224
3
16
290
5455203667702742753
580
Jarek
0,30,44,74,90,120,126,134,156,164,170,200,216,246,260,290
0,260,74,246,200,120,126,134,216,44,290,30,164,156,90,170
3
16
290
581991362272134047
580
Jarek
0,24,60,80,84,104,126,140,150,164,186,206,210,230,266,290
0,266,104,210,230,84,126,140,186,80,290,24,164,150,60,206
3
16
290
7049872471938858473
580
Jarek
0,60,66,74,90,126,134,140,150,156,164,200,216,224,230,290
0,230,134,216,224,126,90,140,156,74,290,60,200,150,66,164
3
16
292
2206032236894340637
584
Jarek
0,6,66,70,72,76,136,142,150,156,216,220,222,226,286,292
0,286,76,222,226,72,150,136,216,70,292,6,142,156,66,220
3
16
296
1922678141706492617
592
Jarek
0,14,30,44,96,110,126,140,156,170,186,200,252,266,282,296
0,266,44,282,200,126,156,110,252,14,296,30,140,186,96,170
3
16
296
3300491071530767057
592
Jarek
0,6,20,26,126,132,144,146,150,152,164,170,270,276,290,296
0,290,26,276,170,132,144,146,270,20,296,6,152,150,126,164
3
16
296
35384185528375907
592
Jarek
0,26,60,84,86,110,126,144,152,170,186,210,212,236,270,296
0,236,86,270,212,144,126,110,210,26,296,60,170,186,84,152
3
16
296
5882949102848933627
592
Jarek
0,30,42,72,110,114,140,144,152,156,182,186,224,254,266,296
0,266,140,186,254,72,114,152,156,110,296,30,182,144,42,224
3
16
296
6515642224149647357
592
Jarek
0,14,30,44,96,110,126,140,156,170,186,200,252,266,282,296
0,266,44,282,200,126,156,110,252,14,296,30,140,186,96,170
3
16
296
6583373692724467757
592
Jarek
0,30,56,66,86,96,122,144,152,174,200,210,230,240,266,296
0,266,86,240,230,96,144,122,210,56,296,30,152,174,66,200
3
16
296
7102186209897030767
592
Jarek
0,42,50,60,92,102,110,144,152,186,194,204,236,246,254,296
0,254,92,246,236,102,144,110,204,50,296,42,152,186,60,194
3
16
296
7404329590632669491
592
Jarek
0,18,38,56,90,108,128,146,150,168,188,206,240,258,278,296
0,278,56,258,206,108,150,128,240,38,296,18,146,168,90,188
3
16
296
839844930382762247
592
Jarek
0,54,60,72,110,114,126,132,164,170,182,186,224,236,242,296
0,242,164,186,236,114,72,170,132,110,296,54,224,126,60,182
3
16
298
117243928896938071
596
Jarek
0,16,30,46,96,112,126,142,156,172,186,202,252,268,282,298
0,268,46,282,202,126,156,112,252,16,298,30,142,186,96,172
3
16
298
2071769060449445971
596
Jarek
0,60,66,70,102,126,130,136,162,168,172,196,228,232,238,298
0,238,130,228,232,126,102,136,168,70,298,60,196,162,66,172
3
16
298
266628508388015611
596
Jarek
0,6,22,28,60,66,82,88,210,216,232,238,270,276,292,298
0,292,28,276,238,66,210,82,270,22,298,6,88,216,60,232
3
16
298
4319730598361536231
596
Jarek
0,18,42,58,60,76,100,118,180,198,222,238,240,256,280,298
0,280,76,240,256,60,180,100,222,58,298,18,118,198,42,238
3
16
298
5797097809523551351
596
Jarek
0,28,72,78,100,106,120,148,150,178,192,198,220,226,270,298
0,226,100,270,220,150,120,106,198,28,298,72,178,192,78,148
3
16
298
6092270314536820741
596
Jarek
0,12,48,60,78,90,126,138,160,172,208,220,238,250,286,298
0,286,172,138,250,60,78,208,126,160,298,12,220,90,48,238
3
16
302
4298579701630856771
604
Jarek
0,12,30,42,110,122,140,150,152,162,180,192,260,272,290,302
0,290,122,192,272,42,150,140,180,110,302,12,152,162,30,260
3
16
302
4996547000548290659
604
Jarek
0,8,42,50,120,128,132,140,162,170,174,182,252,260,294,302
0,260,50,294,182,162,132,128,252,8,302,42,170,174,120,140
3
16
302
6520120848596434049
604
Jarek
0,18,42,50,60,68,92,110,192,210,234,242,252,260,284,302
0,284,68,252,260,60,192,92,234,50,302,18,110,210,42,242
3
16
302
7047060358716199097
604
Jarek
0,26,30,56,66,92,96,122,180,206,210,236,246,272,276,302
0,272,56,276,236,96,180,92,246,26,302,30,122,210,66,206
3
16
302
7214261446565240399
604
Jarek
0,2,48,50,120,122,132,134,168,170,180,182,252,254,300,302
0,254,50,300,182,168,132,122,252,2,302,48,170,180,120,134
3
16
304
2046668965721502079
608
Jarek
0,42,60,70,102,112,130,132,172,174,192,202,234,244,262,304
0,262,112,234,244,102,132,130,192,70,304,42,172,174,60,202
3
16
304
3477773609348442967
608
Jarek
0,30,40,70,84,114,124,150,154,180,190,220,234,264,274,304
0,274,70,264,220,114,150,124,234,40,304,30,154,180,84,190
3
16
304
4009988039455789603
608
Jarek
0,18,60,70,78,88,130,148,156,174,216,226,234,244,286,304
0,286,88,234,244,78,156,130,216,70,304,18,148,174,60,226
3
16
304
5266069044156294013
608
Jarek
0,30,54,70,84,100,124,150,154,180,204,220,234,250,274,304
0,274,100,234,250,84,150,124,204,70,304,30,154,180,54,220
3
16
304
994431411078636463
608
Jarek
0,18,30,48,70,88,100,118,186,204,216,234,256,274,286,304
0,286,88,234,274,48,186,100,216,70,304,18,118,204,30,256
3
16
310
1249673986036931797
620
Jarek
0,6,60,66,84,90,144,150,160,166,220,226,244,250,304,310
0,304,166,150,250,66,84,220,144,160,310,6,226,90,60,244
3
16
310
1810044612251839399
620
Jarek
0,22,30,52,90,112,120,142,168,190,198,220,258,280,288,310
0,280,52,288,220,120,168,112,258,22,310,30,142,198,90,190
3
16
310
6335241952834452079
620
Jarek
0,10,42,48,52,58,90,100,210,220,252,258,262,268,300,310
0,268,52,300,262,90,210,58,258,10,310,42,100,252,48,220
3
16
312
4151299905250340849
624
Jarek
0,12,30,42,102,114,132,144,168,180,198,210,270,282,300,312
0,282,42,300,210,132,168,114,270,12,312,30,144,198,102,180
3
16
314
1133184478443607433
628
Jarek
0,6,20,26,78,84,98,104,210,216,230,236,288,294,308,314
0,308,26,294,236,84,210,98,288,20,314,6,104,216,78,230
3
16
314
1171794689385255863
628
Jarek
0,30,48,78,96,126,140,144,170,174,188,218,236,266,284,314
0,284,170,174,266,78,96,188,144,140,314,30,218,126,48,236
3
16
314
471417195780846839
628
Jarek
0,42,60,80,102,122,132,140,174,182,192,212,234,254,272,314
0,272,122,234,254,102,132,140,192,80,314,42,182,174,60,212
3
16
314
6396245042791103567
628
Jarek
0,42,72,90,110,114,132,152,162,182,200,204,224,242,272,314
0,272,152,204,242,114,90,182,162,110,314,42,224,132,72,200
3
16
314
6405764357124778559
628
Jarek
0,30,42,72,102,132,140,144,170,174,182,212,242,272,284,314
0,284,170,174,272,72,102,182,144,140,314,30,212,132,42,242
3
16
316
102543159231677887
632
Jarek
0,30,46,76,84,114,130,156,160,186,202,232,240,270,286,316
0,286,76,270,232,114,156,130,240,46,316,30,160,186,84,202
3
16
316
437180789975621413
632
Jarek
0,18,28,46,60,78,88,106,210,228,238,256,270,288,298,316
0,298,46,288,256,78,210,88,270,28,316,18,106,228,60,238
3
16
316
471298937749607257
632
Jarek
0,24,30,42,54,66,72,96,220,244,250,262,274,286,292,316
0,292,244,96,286,54,42,250,72,220,316,24,274,66,30,262
3
16
316
656718384132837967
632
Jarek
0,24,70,72,94,96,142,150,166,174,220,222,244,246,292,316
0,292,94,246,244,96,150,142,222,70,316,24,166,174,72,220
3
16
320
1151506862162394281
640
Jarek
0,20,42,62,78,98,120,140,180,200,222,242,258,278,300,320
0,278,62,300,242,120,180,98,258,20,320,42,140,222,78,200
3
16
320
4071507019363633589
640
Jarek
0,42,78,90,110,120,132,152,168,188,200,210,230,242,278,320
0,278,152,210,242,120,90,188,168,110,320,42,230,132,78,200
3
16
320
5710290893708538971
640
Jarek
0,12,30,42,80,92,110,122,198,210,228,240,278,290,308,320
0,308,92,240,290,42,198,110,228,80,320,12,122,210,30,278
3
16
322
1372774196925859801
644
Jarek
0,6,30,36,126,132,156,160,162,166,190,196,286,292,316,322
0,316,166,162,292,36,126,190,156,160,322,6,196,132,30,286
3
16
322
4547410142449237069
644
Jarek
0,30,70,84,100,114,138,154,168,184,208,222,238,252,292,322
0,292,100,252,238,114,138,154,222,70,322,30,184,168,84,208
3
16
322
53517627231758089
644
Jarek
0,30,70,100,108,114,138,144,178,184,208,214,222,252,292,322
0,292,100,252,214,138,114,178,222,70,322,30,208,144,108,184
3
16
322
78083002410887917
644
Jarek
0,46,60,66,106,112,126,150,172,196,210,216,256,262,276,322
0,262,106,276,256,126,150,112,216,46,322,60,172,210,66,196
3
16
326
7227342547726952747
652
Jarek
0,24,66,90,116,120,140,144,182,186,206,210,236,260,302,326
0,302,140,210,260,90,120,182,186,116,326,24,206,144,66,236
3
16
328
1447873525247482441
656
Jarek
0,10,42,52,66,76,108,118,210,220,252,262,276,286,318,328
0,286,52,318,262,108,210,76,276,10,328,42,118,252,66,220
3
16
328
4563371727471654439
656
Jarek
0,24,70,90,94,114,144,160,168,184,214,234,238,258,304,328
0,304,94,258,238,114,144,160,234,70,328,24,184,168,90,214
3
16
328
5088634411611680821
656
Jarek
0,12,40,52,126,138,150,162,166,178,190,202,276,288,316,328
0,316,52,288,202,138,150,166,276,40,328,12,178,162,126,190
3
16
332
3636409035415400477
664
Jarek
0,56,60,90,116,126,146,150,182,186,206,216,242,272,276,332
0,272,116,276,242,150,126,146,216,56,332,60,206,186,90,182
3
16
332
6868687010299798889
664
Jarek
0,2,60,62,102,104,162,164,168,170,228,230,270,272,330,332
0,272,62,330,230,162,168,104,270,2,332,60,164,228,102,170
3
16
334
5561040821824457257
668
Jarek
0,22,30,52,120,142,150,162,172,184,192,214,282,304,312,334
0,304,52,312,214,150,162,142,282,22,334,30,172,192,120,184
3
16
338
2443098533675855513
676
Jarek
0,20,30,50,78,98,108,128,210,230,240,260,288,308,318,338
0,308,50,318,260,108,210,98,288,20,338,30,128,240,78,230
3
16
338
4281439550083215131
676
Jarek
0,8,42,50,78,86,120,128,210,218,252,260,288,296,330,338
0,296,50,330,260,120,210,86,288,8,338,42,128,252,78,218
3
16
338
5095550974094646059
676
Jarek
0,14,30,44,114,128,144,158,180,194,210,224,294,308,324,338
0,308,44,324,224,144,180,128,294,14,338,30,158,210,114,194
3
16
338
5360825217429203801
676
Jarek
0,42,60,96,102,138,140,156,182,198,200,236,242,278,296,338
0,296,182,198,278,102,96,200,156,140,338,42,242,138,60,236
3
16
338
6265903759874222141
676
Jarek
0,56,60,66,116,122,126,156,182,212,216,222,272,278,282,338
0,278,116,282,272,126,156,122,222,56,338,60,182,216,66,212
3
16
338
6807902690803755911
676
Jarek
0,18,38,56,90,108,128,146,192,210,230,248,282,300,320,338
0,320,56,300,248,108,192,128,282,38,338,18,146,210,90,230
3
16
338
7085130922363496141
676
Jarek
0,38,42,80,90,128,132,168,170,206,210,248,258,296,300,338
0,296,80,300,248,132,168,128,258,38,338,42,170,210,90,206
3
16
340
2465521514662779829
680
Jarek
0,60,70,72,130,132,138,142,198,202,208,210,268,270,280,340
0,280,130,270,268,132,138,142,210,70,340,60,202,198,72,208
3
16
340
500155744849852957
680
Jarek
0,24,66,90,120,130,144,154,186,196,210,220,250,274,316,340
0,316,154,210,274,90,120,196,186,130,340,24,220,144,66,250
3
16
340
580958830135976893
680
Jarek
0,30,54,84,100,130,154,156,184,186,210,240,256,286,310,340
0,310,130,240,286,84,156,154,210,100,340,30,184,186,54,256
3
16
344
4340182020207507953
688
Jarek
0,48,50,90,98,138,140,156,188,204,206,246,254,294,296,344
0,296,98,294,254,138,156,140,246,50,344,48,188,204,90,206
3
16
346
3038165674182487351
692
Jarek
0,18,48,66,70,88,118,136,210,228,258,276,280,298,328,346
0,328,88,276,298,66,210,118,258,70,346,18,136,228,48,280
3
16
346
4446407233901388037
692
Jarek
0,6,66,72,84,90,150,156,190,196,256,262,274,280,340,346
0,340,196,156,280,72,84,256,150,190,346,6,262,90,66,274
3
16
346
4651594019187640867
692
Jarek
0,42,70,102,112,132,144,172,174,202,214,234,244,276,304,346
0,304,112,276,244,144,132,172,234,70,346,42,214,174,102,202
3
16
352
2326497647118740959
704
Jarek
0,48,70,114,118,120,162,168,184,190,232,234,238,282,304,352
0,304,118,282,238,162,120,184,234,70,352,48,232,168,114,190
3
16
352
4970537363787854701
704
Jarek
0,30,36,66,126,156,160,162,190,192,196,226,286,316,322,352
0,322,190,192,316,66,126,196,162,160,352,30,226,156,36,286
3
16
352
6142863083364710359
704
Jarek
0,18,70,72,88,90,142,160,192,210,262,264,280,282,334,352
0,334,88,282,280,90,192,142,264,70,352,18,160,210,72,262
3
16
356
1153390150252542617
712
Jarek
0,24,60,84,90,114,150,174,182,206,242,266,272,296,332,356
0,332,206,174,296,84,90,242,150,182,356,24,266,114,60,272
3
16
356
1790884813919768417
712
Jarek
0,36,56,90,92,126,146,174,182,210,230,264,266,300,320,356
0,320,92,300,266,126,174,146,264,56,356,36,182,210,90,230
3
16
356
4088942087717211827
712
Jarek
0,12,60,72,114,126,170,174,182,186,230,242,284,296,344,356
0,344,182,186,296,72,114,230,174,170,356,12,242,126,60,284
3
16
358
2639259096347485153
716
Jarek
0,24,34,58,90,114,124,148,210,234,244,268,300,324,334,358
0,334,58,324,268,114,210,124,300,34,358,24,148,234,90,244
3
16
358
3565892726219917759
716
Jarek
0,54,70,84,124,138,150,154,204,208,220,234,274,288,304,358
0,304,124,288,274,138,150,154,234,70,358,54,208,204,84,220
3
16
358
904317321628148641
716
Jarek
0,6,30,36,126,132,156,162,196,202,226,232,322,328,352,358
0,352,202,162,328,36,126,226,156,196,358,6,232,132,30,322

(*) magic costant=prime*4+magic

Here are the best solution arrived after the ending of the competition:

Task
K-tuple
Diameter
Prime
Magic(*)
Name

(*) magic costant=prime*4+magic

Primes k-tuple

K-Tuple, Magic Squares,

Competition #4

This competition is organized by Макарова Наталия (Natalia Makarova)

Definitions

Definition 1:

A prime k-tuple is a finite collection of values (p + a1, p + a2, p + a3, …, p + ak), where p, p + a1, p + a2, p + a3, …, p + ak are prime numbers, (a1, a2, a3, …, ak) are pattern. Typically the first value in the pattern is 0 and the rest are distinct positive even numbers.

We consider the k-tuple, where p + a1, p + a2, p + a3, …, p + ak are consecutive primes.

Definition 2:

k-tuple (p + a1, p + a2, p + a3, …, p + a [k / 2], p + a [k / 2+1], …, p + a [k-2], p + a [k-1], p + ak) for k even, is called symmetric, if the following condition is satisfied:
a1 + ak = a2 + a[k-1] = a3 + a[k-2] = … = a[k/2] + a[k/2+1]

Example: symmetric 8-tuple

(17 + 0, 17 + 2, 17 + 6, 17 + 12, 17 + 14, 17 + 20, 17 + 24, 17 + 26)

Shortened we write this:

17: 0, 2, 6, 12, 14, 20, 24, 26

Definition 3:

k-tuple (p + a1, p + a2, p + a3, …, p + a [(k-1) / 2], p + a [(k-1) / 2 + 1], p + a [(k-1) / 2 + 2], …, p + a [k-2], p + a [k-1], p + ak) for k odd called symmetric, if the following condition is satisfied:
a1 + ak = a2 + a[k-1] = a3 +a [k-2] =…= a[(k-1)/2] + a[(k-1)/2+2] = 2 a[(k-1)/2+1]

Example: symmetric 5-tuple

18713: 0, 6, 18, 30, 36

Definition 4:

The diameter d of k-tuple is the difference of its largest and smallest elements.

Example: 8-tuple

17: 0, 2, 6, 12, 14, 20, 24, 26

It has a diameter d = 26.

Definition 5:

A pandiagonal magic square is a magic square with the additional property that the broken diagonals, i.e. the diagonals that wrap round at the edges of the square, also add up to the magic constant.

The Contest

In the contest is required to compete for those tasks:

Task 1
Required to find k-tuples with the minimal value p:
for an even k > 24; for odd k > 15.

Example

15-tuple, p=3945769040698829 (minimal)
3945769040698829: 0, 12, 18, 42, 102, 138, 180, 210, 240, 282, 318, 378, 402, 408, 420

Task 2
Required to find k-tuples with a minimal diameter d:
for an even k > 10; for odd k > 13.

Example
8-tuple with a minimal diameter d = 26

17: 0, 2, 6, 12, 14, 20, 24, 26

Task 3
Required to find the 16-tuple, the elements of which it is possible to make pandiagonal magic square of order 4 with magic constant S as: 94615738903617540 < S < 29643562211780078520 Example 16-tuple 

23653934725904299: 0, 12, 22, 34, 48, 60, 70, 82, 90, 102, 112, 124, 138, 150, 160, 172

pandiagonal magic square

23653934725904299+
0  160   60  124
82  102   22  138
112   48  172   12
150   34   90   70

S=94615738903617540

Rule:

Prime numbers can contain no more than 100 digits.

In tasks 1 and 2 k <= 50.

Format of solution

For every task the first line is the number of task you are entering: so 1, 2 or 3. After there are those lines according to the tasks you are inserting.

In tasks 1 and 2

k-tuple is represented as

p: a1, a2, a3, …, ak

Example

18713: 0, 6, 18, 30, 36

In task 3
it is 16-tuple and pandiagonal magic square of order 4, composed of the elements of the tuple.

Example

23653934725904299: 0, 12, 22, 34, 48, 60, 70, 82, 90, 102, 112, 124, 138, 150, 160, 172
0, 160, 60, 124, 82, 102, 22, 138, 112, 48, 172, 12, 150, 34, 90, 70

Scoring

The contestant receives one point for every solution to tasks 1 and 3.
For each k in task 2 one point counted towards only those participants, who will have a minimum diameter d.

The Prize

If two or more contestants have equal number of points, the winner will be the entrant who submitted solutions ahead of other contestants.
The winner will receive a prize of 5,000 rubles.
If the winner is not from Russia, the prize will be paid in US dollars at the official exchange rate on the day ending of the contest.

Thanks

We thanks  Wolfram Alpha as we use their API for testing the primality of the given big numbers.

Links

[1] https://en.wikipedia.org/wiki/Prime_k-tuple
[2] https://en.wikipedia.org/wiki/Pandiagonal_magic_square
[3] http://oeis.org/A256234
[4] http://oeis.org/A081235
[5] http://oeis.org/A055380
[6] http://oeis.org/A055382
[7] http://oeis.org/A175309
[8] http://www.primepuzzles.net/problems/prob_060.htm
[9] http://dxdy.ru/topic93581.html
[10] http://dxdy.ru/topic87170.html

Result for Ultra Magic Squares of prime numbers

Magic Squares, Result, ,

The competition Ultra Magic Squares of prime numbers is over.

Thanks to all people that have partecipate.

The result is in this table:

Task
Dim.
User
Magic
Assoc.
Result
0
7
primesmagicgames
4613
1318
227 617 677 431 1217 1307 137 1259 827 1061 509 521 167 269 347 929 1187 17 557 719 857 89 479 29 659 1289 839 1229 461 599 761 1301 131 389 971 1049 1151 797 809 257 491 59 1181 11 101 887 641 701 1091
0
8
Natalia Makarova
2040
510
241 199 409 467 47 79 359 239
421 137 7 53 487 179 317 439
31 281 347 353 227 277 127 397
449 197 109 379 491 337 11 67
443 499 173 19 131 401 313 61
113 383 233 283 157 163 229 479
71 193 331 23 457 503 373 89
271 151 431 463 43 101 311 269
0
9
Natalia Makarova
13059
2902
2843 149 1973 2039 971 1031 2141 293 1619
2063 563 1811 113 2549 1601 2633 1721 5
2393 503 1613 2381 1193 41 2411 101 2423
173 2711 2879 773 1583 1493 461 443 2543
569 83 821 311 1451 2591 2081 2819 2333
359 2459 2441 1409 1319 2129 23 191 2729
479 2801 491 2861 1709 521 1289 2399 509
2897 1181 269 1301 353 2789 1091 2339 839
1283 2609 761 1871 1931 863 929 2753 59
0
10
Natalia Makarova
46150
9230
9133 2017 1069 1669 3583 4999 8629 1489 6343 7219 5209 4219 5101 6793 43 6841 7951 2683 5557 1753 7603 7369 6883 8059 8863 919 1471 769 4111 103 163 8179 4723 4243 4663 5869 1741 6553 1723 8293 2833 4051 709 1021 7177 5701 1993 6991 8101 7573 1657 1129 2239 7237 3529 2053 8209 8521 5179 6397 937 7507 2677 7489 3361 4567 4987 4507 1051 9067 9127 5119 8461 7759 8311 367 1171 2347 1861 1627 7477 3673 6547 1279 2389 9187 2437 4129 5011 4021 2011 2887 7741 601 4231 5647 7561 8161 7213 97

Here are the best solution arrived after the ending of the competition:

Task
Dim.
User
Magic
Assoc.
Result

Ultra Magic Squares of prime numbers

Magic Squares

Competition #3

This competition is organized by Макарова Наталия (Natalia Makarova)

Definitions

Definition 1:

Magic square is called ultra magic square, if it is both associative (center symmetric) and pandiagonal.

Ultra magic squares exist for orders n > 4.

The Contest

In the contest is required to built ultra magic squares of order 7 – 16 of distinct primes.

Example

n = 5
 
 113 1151 1229  911  101
 839  521   41 1013 1091
 941  953  701  449  461
 311  389 1361  881  563
1301  491  173  251 1289

S=3505

This is the minimal solution for n = 5.

Known as a minimal solution for n = 6 (author M. Alekseyev)

103  59 163 233 139 293
229 257 307 131  13  53
283  17  67 173 181 269
 61 149 157 263 313  47
277 317 199  23  73 101
 37 191  97 167 271 227

S=990

Rule:

For each order n = 7 – 16 you can imagine several solutions with magic constants S1 <S2 <S3 …

Known solutions:

n = 7, S = 4613 (author N. Makarova)
n = 8, S = 2640 (author N. Makarova)
n = 9, S = 24237 (author A. Chernov)

Contestant shall not be deemed a winner, if he would submit only solutions with known magic constants

Format of solution

The solution is represented in the form:

n: a (1), a (2), a (3), …, a (n^2)

Example:

5:113,1151,1229,911,101,839,521,41,1013,1091,941,953,701,449,461,311,389,1361,881,563,1301,491,173,251,1289

Scoring

Contestant receives for every n the score: Smin/S, where

Smin – the minimal magic constant of solution in the contest;

S –  the minimal magic constant of solution by contestant.

 

The Prize

Winner receives a prize of 3000 rubles.

If the winner is not from Russia, the prize will be paid in $USA at the official rate on the day of end the contest.

Organizers N. Makarova and S. Tognon can participate in the contest, but does not receive the prize in case of winning.