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On this web page
http://www.magic-squares.net/primesqr.htmyou can see the concentric magic square of order 13 composed of prime numbers
1153 8923 1093 9127 1327 9277 1063 9133 9661 1693 991 8887 8353 9967 8161 3253 2857 6823 2143 4447 8821 8713 8317 3001 3271 907 1831 8167 4093 7561 3631 3457 7573 3907 7411 3967 7333 2707 9043 9907 7687 7237 6367 4597 4723 6577 4513 4831 6451 3637 3187 967 1723 7753 2347 4603 5527 4993 5641 6073 4951 6271 8527 3121 9151 9421 2293 6763 4663 4657 9007 1861 5443 6217 6211 4111 8581 1453 2011 2683 6871 6547 5227 1873 5437 9001 5647 4327 4003 8191 8863 9403 8761 3877 4783 5851 5431 9013 1867 5023 6091 6997 2113 1471 1531 2137 7177 6673 5923 5881 5233 4801 5347 4201 3697 8737 9343 9643 2251 7027 4423 6277 6151 4297 6361 6043 4507 3847 8623 1231 1783 2311 3541 3313 7243 7417 3301 6967 3463 6907 6781 8563 9091 9787 7603 7621 8017 4051 8731 6427 2053 2161 2557 7873 2713 1087 2521 1951 9781 1747 9547 1597 9811 1741 1213 9181 9883 1987 9721S=70681
I got some interesting results on this issue. I’ll show these results.
And I talk about the unresolved issues.We consider concentric magic squares composed of prime numbers.
1. I found a minimal concentric magic square of order 5 with magic constant S=1255.
2. I found a concentric magic square of order 6 with magic constant S=630.
Maybe it’s not a minimal solution.
I invite colleagues to investigate this matter.To be continued…
5. I found a minimal concentric magic square of order 9 with magic constant S=12249
6. I found a minimal concentric magic square of order 10 with magic constant S=4200
Maybe it’s not a minimal solution.
Here we have an unsolved problem.
In sequence OEIS
https://oeis.org/A188537you see: a(10) <= 4950
I found an associative square of order 10 with magic constant S = 4950
593 977 757 181 331 307 569 281 193 761 631 541 953 83 677 131 257 419 641 617 673 127 383 137 239 727 719 433 911 601 937 113 103 467 983 827 43 167 499 811 839 547 23 971 109 347 251 919 61 883 107 929 71 739 643 881 19 967 443 151 179 491 823 947 163 7 523 887 877 53 389 79 557 271 263 751 853 607 863 317 373 349 571 733 859 313 907 37 449 359 229 797 709 421 683 659 809 233 13 397K=990, S=4950
But I have not found a minimal associative square of order 10 with magic constant
S = 4200.
Can anyone solve this problem?To be continued…
7. I found a minimal concentric magic square of order 11 with magic constant S=26521
8. I found a concentric magic square of order 12 with magic constant S=8820
Maybe it’s not a minimal solution.
There are minimal associative squares of orders 11, 12.
See https://oeis.org/A188537a(11) = 26521, a(12)=8820
I found these solutions:
n=11
4079 4673 401 3911 1151 2111 1619 1229 4583 1601 1163 2381 1361 4733 4253 941 1523 1823 3803 809 4721 173 563 1499 1373 179 4001 3371 2579 2213 3539 2741 4463 3389 1979 431 1493 2069 4049 3413 23 1913 4229 3533 2819 3209 1571 3851 2399 3719 449 4139 5 1811 2549 1061 4793 4751 131 1733 2411 3089 4691 71 29 3761 2273 3011 4817 683 4373 1103 2423 971 3251 1613 2003 1289 593 2909 4799 1409 773 2753 3329 4391 2843 1433 359 2081 1283 2609 2243 1451 821 4643 3449 3323 4259 4649 101 4013 1019 2999 3299 3881 569 89 3461 2441 3659 3221 239 3593 3203 2711 3671 911 4421 149 743K=4822, S=26521
n=12
241 1061 11 307 523 317 1429 809 719 1021 1291 1091 1277 1237 487 1039 37 1427 659 181 1049 503 277 647 541 587 727 907 1187 163 499 1013 1399 643 1051 103 31 877 1297 1069 797 1451 761 461 829 211 683 353 439 191 1231 701 149 1303 1097 1123 857 853 607 269 1381 17 839 1087 1373 151 367 479 257 1361 61 1447 23 1409 109 1213 991 1103 1319 97 383 631 1453 89 1201 863 617 613 347 373 167 1321 769 239 1279 1031 1117 787 1259 641 1009 709 19 673 401 173 593 1439 1367 419 827 71 457 971 1307 283 563 743 883 929 823 1193 967 421 1289 811 43 1433 431 983 233 193 379 179 449 751 661 41 1153 947 1163 1459 409 1229K=1470, S=8820
To be continued…
9. I found a minimal concentric magic square of order 13 with magic constant S=49439
10. I found a concentric magic square of order 14 with magic constant S=16170
Maybe it’s not a minimal solution.
Here we have an unsolved problem.
In sequence OEIS
https://oeis.org/A188537
you see: a(13) <= 50453I found an associative square of order 13 with magic constant S = 50453
5 7673 1409 5939 659 7193 5861 2789 7589 1061 5153 179 4943 5099 59 7283 5021 3533 3083 683 6263 5519 4391 1979 431 7109 3911 71 6581 5231 113 6899 5009 6599 743 2969 5693 3011 3623 3089 2411 6269 1559 1913 4919 2381 7499 641 4001 5813 6659 3299 7001 3389 5651 2459 6389 1193 3833 6311 929 3413 4271 4643 971 5051 281 719 6053 5171 7253 4091 4721 4799 7529 1433 1733 1619 1289 6653 311 983 6323 7523 3881 239 1439 6779 7451 1109 6473 6143 6029 6329 233 2963 3041 3671 509 2591 1709 7043 7481 2711 6791 3119 3491 4349 6833 1451 3929 6569 1373 5303 2111 4373 761 4463 1103 1949 3761 7121 263 5381 2843 5849 6203 1493 5351 4673 4139 4751 2069 4793 7019 1163 2753 863 7649 2531 1181 7691 3851 653 7331 5783 3371 2243 1499 7079 4679 4229 2741 479 7703 2663 2819 7583 2609 6701 173 4973 1901 569 7103 1823 6353 89 7757K=7762, S=50453
But I have not found a minimal associative square of order 13 with magic constant
S = 49439.
Can anyone solve this problem?To be continued…
11. I found a minimal concentric magic square of order 15 with magic constant S=74595
12. I found a concentric magic square of order 16 with magic constant S=21840
Maybe it’s not a minimal solution.
In sequence OEIS
https://oeis.org/A188537
you see: a(15)=74595, a(16)<=25200I found a minimal associative square of order 15 with magic constant S=74595
6983 9137 1277 5807 5273 2357 7109 2153 8573 9437 5309 107 1559 7487 2027 8963 773 359 6779 4133 9497 3557 947 7523 4289 4049 5939 4253 3593 9941 3083 5573 653 5399 8753 5717 2039 7529 1997 2069 9923 9629 5867 2447 3917 9203 8837 2063 2693 1163 977 8423 6857 6977 3833 7193 719 3617 4937 7103 3623 3119 8849 5297 9059 2243 2399 2339 479 8363 9857 4943 3659 3677 6689 1709 569 9719 2903 9929 7247 3209 1877 3347 1283 8933 5147 4793 8447 5483 9677 467 8537 2129 9689 179 8627 2999 2927 3803 59 7673 8039 2663 7127 4019 9749 5303 8123 2297 9833 1433 4973 8513 113 7649 1823 4643 197 5927 2819 7283 1907 2273 9887 6143 7019 6947 1319 9767 257 7817 1409 9479 269 4463 1499 5153 4799 1013 8663 6599 8069 6737 2699 17 7043 227 9377 8237 3257 6269 6287 5003 89 1583 9467 7607 7547 7703 887 4649 1097 6827 6323 2843 5009 6329 9227 2753 6113 2969 3089 1523 8969 8783 7253 7883 1109 743 6029 7499 4079 317 23 7877 7949 2417 7907 4229 1193 4547 9293 4373 6863 5 6353 5693 4007 5897 5657 2423 8999 6389 449 5813 3167 9587 9173 983 7919 2459 8387 9839 4637 509 1373 7793 2837 7589 4673 4139 8669 809 2963K=9946, S=74595
For n = 16 we have unsolved problems.
I found a associative square of order 16 with magic constant S=252001087 619 2089 607 1249 677 397 2659 2857 2213 1543 151 2053 3001 2797 1201 1481 1063 421 2801 271 2843 773 1549 2027 313 1013 2593 2731 2767 1697 857 179 1997 3121 2099 683 3019 1483 2221 907 601 223 2741 1571 2749 1163 443 2953 3089 2551 863 2557 1597 181 1367 1627 1409 109 1823 1723 113 461 2777 2267 1777 1933 1493 1721 101 2477 479 769 2719 2081 233 1019 839 2609 2683 193 3119 127 1913 2297 1873 751 139 709 971 881 3083 1831 1871 2503 2939 2711 1283 1289 1583 2803 947 997 1151 1789 241 3137 2887 811 89 2791 691 761 2693 2239 463 2663 1009 2579 2423 1303 1171 3079 733 83 2341 1619 41 3109 1531 809 3067 2417 71 1979 1847 727 571 2141 487 2687 911 457 2389 2459 359 3061 2339 263 13 2909 1361 1999 2153 2203 347 1567 1861 1867 439 211 647 1279 1319 67 2269 2179 2441 3011 2399 1277 853 1237 3023 31 2957 467 541 2311 2131 2917 1069 431 2381 2671 673 3049 1429 1657 1217 1373 883 373 2689 3037 1427 1327 3041 1741 1523 1783 2969 1553 593 2287 599 61 197 2707 1987 401 1579 409 2927 2549 2243 929 1667 131 2467 1051 29 1153 2971 2293 1453 383 419 557 2137 2837 1123 1601 2377 307 2879 349 2729 2087 1669 1949 353 149 1097 2999 1607 937 293 491 2753 2473 1901 2543 1061 2531 2063K=3150, S=25200
But I have not found a minimal associative square of order 16 with magic constant
S = 21840.
Can anyone solve this problem?To be continued…
13. I found a minimal concentric magic square of order 17 with magic constant S=128197
6569 6581 6653 6659 6719 6911 6959 6971 7001 7013 7229 7253 7433 7499 7559 11069 12119 2801 5279 5333 5393 5531 5651 5669 5711 5741 5801 5843 5861 12539 12809 12953 13001 12281 2843 4079 4133 4229 4283 4349 4373 4391 4451 4481 9491 13103 13463 13523 13763 11003 12239 2879 4871 4493 3449 3461 3533 3593 3659 3671 6113 13709 13781 13931 14051 10589 10211 12203 3761 4889 4523 2969 2339 2393 2411 2423 2441 12263 14489 14519 14591 12113 10559 10193 11321 6329 4919 4583 3011 1823 5 191 269 8219 14633 14699 14771 13259 12071 10499 10163 8753 8609 4931 4691 3041 1931 359 839 929 6053 14831 15053 14723 13151 12041 10391 10151 6473 8693 4943 4751 3251 4649 8291 7043 13649 113 8861 8039 6791 10433 11831 10331 10139 6389 8951 5003 4793 11279 13859 14561 14843 2753 7541 12329 239 521 1223 3803 10289 10079 6131 9203 9929 12143 13499 14033 14621 14951 6221 14969 1433 131 461 1049 1583 2939 5153 5879 9851 11783 12983 13721 14321 14639 29 14153 9029 251 14243 443 761 1361 2099 3299 5231 10061 12473 13043 13799 14423 311 14891 14813 6863 449 383 15077 659 1283 2039 2609 5021 11093 12491 13109 13901 491 12689 12671 12659 12641 2819 593 563 12743 1181 1973 2591 3989 11159 12503 13469 1031 11621 11549 11489 11423 11411 8969 1373 1301 1151 11633 1613 2579 3923 11171 12941 1319 10853 10799 10733 10709 10691 10631 10601 5591 1979 1619 1559 10949 2141 3911 11261 2081 9749 9689 9551 9431 9413 9371 9341 9281 9239 9221 2543 2273 2129 9803 3821 2963 8501 8429 8423 8363 8171 8123 8111 8081 8069 7853 7829 7649 7583 7523 4013 8513http://s29.postimg.org/yz4v4037r/concentr17_Min.png
14. I found a concentric magic square of order 18 with magic constant S=35910
991 1019 1021 1399 1433 1607 1609 1613 1619 1657 1693 1877 2269 3023 3167 3181 3359 3373 1901 1439 1447 1451 1459 1523 1531 1543 1549 1553 1567 1579 1601 3347 3371 3433 3527 2089 1907 1051 1187 1193 1201 1213 1223 1237 1249 1277 1291 2503 3533 3593 3607 3623 2939 2083 1951 1171 1063 827 853 881 907 911 929 941 3019 3631 3659 3673 3709 2927 2819 2039 1973 1297 1087 761 677 683 691 719 733 1327 3739 3761 3767 3853 3229 2903 2693 2017 1997 1301 1093 769 647 499 521 523 541 2281 3851 3863 3881 3343 3221 2897 2689 1993 2003 1307 1103 773 659 443 59 61 113 3907 3911 3919 3547 3331 3217 2887 2683 1987 2111 1319 1129 787 661 449 211 23 3947 3943 67 3779 3541 3329 3203 2861 2671 1879 2143 1373 1153 1597 2293 3037 3823 1303 2707 2711 1259 167 953 1697 2393 2837 2617 1847 2179 1381 2011 3571 3697 3793 3889 2731 1279 1283 2687 101 197 293 419 1979 2609 1811 2203 2957 3511 3617 3719 3797 3917 3923 47 43 3967 73 193 271 373 479 1033 1787 2207 3169 3529 3637 3727 3833 71 3929 3877 83 79 3931 157 263 353 461 821 1783 2213 3389 3557 3643 3733 109 3469 3467 3449 1709 139 127 3491 257 347 433 601 1777 2243 3391 3559 3677 137 3307 3299 3271 3257 2663 251 229 223 3313 313 431 599 1747 2267 3413 3581 281 3137 3109 3083 3079 3061 3049 971 359 331 317 3163 409 577 1723 3251 3499 367 2797 2789 2777 2767 2753 2741 2713 2699 1487 457 397 383 2803 491 739 1753 463 2543 2539 2531 2467 2459 2447 2441 2437 2423 2411 2389 643 619 557 2551 2237 617 2971 2969 2591 2557 2383 2381 2377 2371 2333 2297 2113 1721 967 823 809 631 2999Maybe it’s not a minimal solution.
I have not found a minimal associative squares of order 17, 18 for sequence OEIS https://oeis.org/A188537
I ask colleagues to solve this problem.To be continued…
I found a minimal associative square of order 18
929 1303 419 1901 2143 3343 599 2237 3169 3499 3251 3947 1373 1033 79 3779 193 2713 3527 643 1567 2797 1301 331 3917 1433 601 3433 139 1619 373 1553 3823 3889 3877 1087 1283 3919 3137 3257 991 809 1213 263 2281 359 2269 2447 3547 2381 1697 787 2803 2467 229 3923 2609 2693 3373 3299 1171 3637 971 1237 397 2003 577 2179 3557 223 2393 1439 3491 1223 313 2699 2731 3167 1777 3037 457 2539 2927 2543 1153 257 2683 3359 23 1531 461 1129 3217 293 3931 449 1327 2017 3739 3023 1997 317 1487 3623 1879 1021 2441 3559 1607 1103 3163 3271 3853 1693 659 1201 881 383 911 197 3719 3907 3943 1723 1613 2083 2129 3229 3313 3371 1601 2039 1019 661 683 1249 3469 541 2207 3929 3863 1319 941 347 3581 523 2243 1399 1051 1093 3511 2333 3709 769 1459 1579 1787 2113 157 3881 2711 2011 1979 1279 109 3833 1877 2203 2411 2531 3221 281 1657 479 2897 2939 2591 1747 3467 409 3643 3049 2671 127 61 1783 3449 521 2741 3307 3329 2971 1951 2389 619 677 761 1861 1907 2377 2267 47 83 271 3793 3079 3607 3109 2789 3331 2297 137 719 827 2887 2383 431 1549 2969 2111 367 2503 3673 1993 967 251 1973 2663 3541 59 3697 773 2861 3529 2459 3967 631 1307 3733 2837 1447 1063 1451 3533 953 2213 823 1259 1291 3677 2767 499 2551 1597 3767 433 1811 3413 1987 3593 2753 3019 353 2819 691 617 1297 1381 67 3761 1523 1187 3203 2293 1609 443 1543 1721 3631 1709 3727 2777 3181 2999 733 853 71 2707 2903 113 101 167 2437 3617 2371 3851 557 3389 2557 73 3659 2689 1193 2423 3347 463 1277 3797 211 3911 2957 2617 43 739 491 821 1753 3391 647 1847 2089 3571 2687 3061K=3990, S=35910
I found a minimal associative square of order 17
3761 13499 8501 2843 2879 1433 4451 4373 12143 5021 12659 9551 8819 6791 9281 9239 12953 8171 29 7583 6569 6971 14969 9689 9029 10691 12941 5669 8363 13799 2591 4871 4649 1613 10151 11261 3911 491 10139 6659 14723 461 8081 8783 5879 10601 1049 8069 7829 13151 6959 13901 3251 1301 3011 11633 7649 13931 13463 2969 2753 12671 251 11423 3989 14321 9341 2339 3671 14639 9851 383 10289 5591 12503 7523 1823 12281 8609 3299 8969 1973 8039 8951 9803 4751 14891 13721 521 4691 2963 1031 13001 2039 14423 14153 13859 3803 4079 3461 5861 10949 10079 1319 12983 12041 563 12641 6221 12809 2393 13709 4493 4919 1979 4229 10193 5153 12473 14489 4013 4283 14243 5711 269 14771 6863 449 3593 15077 2543 3923 7229 11549 4349 14843 10499 5333 14951 6389 8429 8753 9431 13523 7541 1559 5651 6329 6653 8693 131 9749 4583 239 10733 3533 7853 11159 12539 5 11489 14633 8219 311 14813 9371 839 10799 11069 593 2609 9929 4889 10853 13103 10163 10589 1373 12689 2273 8861 2441 14519 3041 2099 13763 5003 4133 9221 11621 11003 11279 1223 929 659 13043 2081 14051 12119 10391 14561 1361 191 10331 5279 6131 7043 13109 6113 11783 6473 2801 13259 7559 2579 9491 4793 14699 5231 443 11411 12743 5741 761 11093 3659 14831 2411 12329 12113 1619 1151 7433 3449 12071 13781 11831 1181 8123 1931 7253 7013 14033 4481 9203 6299 7001 14621 359 8423 4943 14591 11171 3821 4931 13469 10433 10211 12491 1283 6719 9413 2141 4391 6053 5393 113 8111 8513 7499 15053 6911 2129 5843 5801 8291 6263 5531 2423 10061 2939 10709 10631 13649 12203 12239 6581 1583 11321K=15082, S=128197
To be continued…
15. I found a minimal concentric magic square of order 19 with magic constant S=191159
8699 8849 8951 8963 9029 9173 9239 9323 9413 9431 9521 9533 9689 9833 9851 9929 12203 13001 16529 3191 7211 7331 7433 7451 7481 7583 7643 7649 7793 7841 7853 7883 9791 17159 17573 17579 17783 16931 3491 4493 5231 6323 6359 6473 6569 6599 6653 6659 6701 6791 13163 18119 18251 18503 18521 15629 16631 4349 4649 4973 5309 5351 5381 5399 5483 5501 5531 5573 12893 18413 18539 18671 18749 15149 15473 15773 7079 4679 5021 5711 3803 3821 3929 4019 4049 4241 10499 18959 19031 19139 19181 14411 15101 15443 13043 8111 4721 5039 5801 3221 2633 2729 2789 2801 2939 18143 19403 19553 19559 16901 14321 15083 15401 12011 8219 4793 5153 5879 3251 2063 5 59 149 10631 19841 19853 19889 18059 16871 14243 14969 15329 11903 10181 7001 5171 6089 3299 2081 359 821 863 8861 19739 20021 19763 18041 16823 14033 14951 13121 9941 11549 7013 6863 6113 3359 6263 11369 9341 18311 173 11699 10781 8753 13859 16763 14009 13259 13109 8573 11579 7019 6971 6221 17099 18803 19379 19991 3449 10061 16673 131 743 1319 3023 13901 13151 13103 8543 11621 15641 15233 14843 18713 19211 19421 20051 8423 19949 1811 71 701 911 1409 5279 4889 4481 8501 11831 16421 17681 17981 18899 19463 19661 101 19259 11261 383 19301 461 659 1223 2141 2441 3701 8291 12041 16451 17729 18233 19013 19469 233 20063 19973 9491 281 269 20117 653 1109 1889 2393 3671 8081 12113 16661 17909 18401 19073 563 17393 17333 17321 17183 1979 719 569 17489 1049 1721 2213 3461 8009 15671 17033 18149 18839 941 16301 16193 16103 16073 15881 9623 1163 1091 983 16319 1283 1973 3089 4451 15731 17123 18191 1373 14771 14741 14723 14639 14621 14591 14549 7229 1709 1583 1451 14813 1931 2999 4391 15749 17789 1601 13799 13763 13649 13553 13523 13469 13463 13421 13331 6959 2003 1871 1619 14891 2333 4373 16361 2339 12791 12689 12671 12641 12539 12479 12473 12329 12281 12269 12239 10331 2963 2549 2543 12911 3761 3593 11273 11171 11159 11093 10949 10883 10799 10709 10691 10601 10589 10433 10289 10271 10193 7919 7121 11423http://s9.postimg.org/5fcsqcv9r/Concentr19_Nin.png
To be continued…
16. I found a concentric magic square of order 20 with magic constant S=54600
1361 1367 1381 1409 1433 1439 1447 2543 2551 2557 2609 2617 2657 2659 4133 4139 4229 4597 4673 4799 1103 2273 2293 2297 2341 2351 2377 2381 2393 2399 2411 2423 2437 2441 2459 2503 3989 4651 4721 4357 1123 1733 1759 1901 1931 1933 1949 1993 1997 1999 2003 2011 2027 2339 4919 4951 4969 4999 3727 4337 1163 1741 2053 1783 1787 1789 1801 1823 1847 1867 1877 1879 1889 4691 5051 5059 5077 3407 3719 4297 1171 2141 2069 1493 1543 1549 1553 1571 1579 1583 1597 1607 4733 5113 5153 5179 3967 3391 3319 4289 1187 2153 2087 1531 1627 1039 1051 1063 1069 1087 1097 5107 5227 5237 5323 3833 3929 3373 3307 4273 1201 2161 2089 1609 1637 1283 557 571 599 643 3547 5279 5297 5347 4177 3823 3851 3371 3299 4259 1217 2203 2099 1613 1657 1301 467 19 41 157 5333 5413 5417 4993 4159 3803 3847 3361 3257 4243 1229 2207 2113 1693 1663 1303 487 251 11 5443 5437 29 5209 4973 4157 3797 3767 3347 3253 4231 2663 2239 2129 1699 1667 1453 4441 5309 79 5387 5393 61 151 1019 4007 3793 3761 3331 3221 2797 4153 2243 2131 1721 2269 5167 5233 5351 5399 67 73 5381 109 227 293 3191 3739 3329 3217 1307 4201 2251 2137 4621 5081 5189 5261 5407 5431 23 17 5449 53 199 271 379 839 3323 3209 1259 4447 2521 3089 5003 5087 5197 5281 43 5419 5303 127 47 5441 179 263 373 457 2371 2939 1013 4451 4001 4783 5011 5101 5231 113 4889 4861 4817 1913 181 163 4903 229 359 449 677 1459 1009 4463 4211 4787 5021 5147 137 4409 4397 4391 4373 4363 353 233 223 4421 313 439 673 1249 997 4639 4751 4937 5039 281 3911 3907 3889 3881 3877 3863 3853 727 347 307 3917 421 523 709 821 4663 4759 4957 383 3673 3671 3659 3637 3613 3593 3583 3581 3571 769 409 401 3677 503 701 797 4703 4813 461 3559 3529 3527 3511 3467 3463 3461 3457 3449 3433 3121 541 509 491 3701 647 757 4801 739 3167 3163 3119 3109 3083 3079 3067 3061 3049 3037 3023 3019 3001 2957 1471 809 3187 659 661 4093 4079 4051 4027 4021 4013 2917 2909 2903 2851 2843 2803 2801 1327 1321 1231 863 787 4099http://s23.postimg.org/j21uvtw4r/concentr20.png
Maybe it’s not a minimal solution.
I found also a minimal associative square of order 20 with magic constant S=54600
4523 1327 2609 3581 67 199 3727 787 4649 5107 127 4937 5381 3361 11 1283 4969 4651 113 3191 2503 977 4813 1597 1069 5197 1231 3547 1993 5237 401 1741 5417 5281 4363 677 2767 701 1901 3187 1087 3037 2069 1123 4441 157 5231 5437 4957 2789 1847 2011 359 4357 1949 1999 1933 5297 73 4447 383 347 4451 2341 2203 1627 3929 2137 1801 2711 1259 3847 1307 2843 5051 4597 5419 2683 2207 3457 4507 1187 1171 4297 4639 1609 4861 1867 4259 3167 4463 4013 3739 1877 2161 1543 3001 487 1303 449 5441 5309 1439 1931 2239 2243 1229 509 2719 2339 4799 1553 1699 3089 2551 4583 947 181 4999 4801 757 4493 1759 2393 2437 4007 1783 4621 61 1249 5443 1571 1471 4051 2053 3853 1063 4001 2801 4733 769 1493 1657 4663 2129 3877 4243 2377 2113 1667 3797 5167 5003 3371 5039 739 3671 2153 439 233 5081 911 4889 4751 2543 3209 4139 1039 227 673 5413 271 557 2707 307 3673 2131 2297 5179 4603 5431 4093 1693 2657 1823 4409 1997 3079 2441 5407 2797 109 467 1433 3049 643 3433 4099 4159 1381 4079 1301 1361 2027 4817 2411 4027 4993 5351 2663 53 3019 2381 3463 1051 3637 2803 3767 1367 29 857 281 3163 3329 1787 5153 2753 4903 5189 47 4787 5233 4421 1321 2251 2917 709 571 4549 379 5227 5021 3307 1789 4721 421 2089 457 293 1663 3793 3347 3083 1217 1583 3331 797 3803 3967 4691 727 2659 1459 4397 1607 3407 1409 3989 3889 17 4211 5399 839 3677 1453 3023 3067 3701 967 4703 659 461 5279 4513 877 2909 2371 3761 3907 661 3121 2741 4951 4231 3217 3221 3529 4021 151 19 5011 4157 4973 2459 3917 3299 3583 1721 1447 997 2293 1201 3593 599 3851 821 1163 4289 4273 953 2003 3253 2777 41 863 409 2617 4153 1613 4201 2749 3659 3323 1531 3833 3257 3119 1009 5113 5077 1013 5387 163 3527 3461 3511 1103 5101 3449 3613 2671 503 23 229 5303 1019 4337 3391 2423 4373 2273 3559 4759 2693 4783 1097 179 43 3719 5059 223 3467 1913 4229 263 4391 3863 647 4483 2957 2269 5347 809 491 4177 5449 2099 79 523 5333 353 811 4673 1733 5261 5393 1879 2851 4133 937K=5460, S=54600
I have not found minimal associative square of order 19 with magic constant S = 191159.
I have only associative square of order 19 with magic constant
S = 193363.6133 6547 14401 2503 11071 2647 16111 1213 20143 8971 18691 883 15361 1117 19501 14713 577 13513 19267 7591 8053 3391 3907 2137 19597 5167 14947 16993 14923 13 14431 20047 9181 3607 12043 11197 6271 19867 17293 5113 3163 8887 20023 4363 19183 19681 6673 17041 5527 3613 13873 4021 1831 8803 5503 10111 18661 14731 16411 241 20347 11113 5077 12163 17467 7867 8101 7177 17047 18553 1381 3511 9601 8467 13297 811 5791 17383 13003 18133 1597 463 18061 11677 2131 5557 19447 17317 2803 3433 2617 18493 13033 12637 9787 15331 14011 2593 12613 67 2053 19753 15073 8317 7573 4441 9697 19813 11593 7933 19531 2857 8623 11491 2311 9397 16657 20173 6763 14281 17053 16477 4447 8521 11617 14107 15451 5323 2341 13693 1987 11827 937 15493 10723 2143 8011 14143 7753 11287 3217 17623 5821 877 17683 20161 15787 15733 5851 283 11311 9463 7207 751 1567 9001 7963 19387 17977 2383 5233 9643 13933 14347 16057 17401 13327 9901 16981 1873 8431 17011 11251 13903 19231 5701 3793 15541 17167 2473 10177 17881 3187 4813 16561 14653 1123 6451 9103 3343 11923 18481 3373 10453 7027 2953 4297 6007 6421 10711 15121 17971 2377 967 12391 11353 18787 19603 13147 10891 9043 20071 14503 4621 4567 193 2671 19477 14533 2731 17137 9067 12601 6211 12343 18211 9631 4861 19417 8527 18367 6661 18013 15031 4903 6247 8737 11833 15907 3877 3301 6073 13591 181 3697 10957 18043 8863 11731 17497 823 12421 8761 541 10657 15913 12781 12037 5281 601 18301 20287 7741 17761 6343 5023 10567 7717 7321 1861 17737 16921 17551 3037 907 14797 18223 8677 2293 19891 18757 2221 7351 2971 14563 19543 7057 11887 10753 16843 18973 1801 3307 13177 12253 12487 2887 8191 15277 9241 7 20113 3943 5623 1693 10243 14851 11551 18523 16333 6481 16741 14827 3313 13681 673 1171 15991 331 11467 17191 15241 3061 487 14083 9157 8311 16747 11173 307 5923 20341 5431 3361 5407 15187 757 18217 16447 16963 12301 12763 1087 6841 19777 5641 853 19237 4993 19471 1663 11383 211 19141 4243 17707 9283 17851 5953 13807 14221K=20354, S=193363
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