Magic Constants Involving Prime Numbers

I recently turned 27180 days old and one of the properties of this number qualifies it for membership in OEIS A192087:

A192087

Potential magic constants of a 10 X 10 magic square composed of consecutive primes.

The members of this sequence are (permalink):

2862, 3092, 3500, 4222, 4780, 5608, 7124, 10126, 10198, 11212, 11426, 12140, 12212, 12284, 12356, 12428, 12714, 12854, 12924, 15270, 16252, 16476, 18594, 18672, 18750, 18828, 19214, 20764, 21150, 23752, 24214, 24598, 24828, 27180, 27342, 27424, 27916, 28666, 29406, 29568

The OEIS comments are:

For a 10 X 10 magic square composed of 100 consecutive primes, the sum of these primes must be a multiple of 20. This sequence consists of even integers equal the sum of 100 consecutive primes divided by 10. It is not known whether each such set of consecutive primes can be arranged into a 10 X 10 magic square but it looks plausible. Actual magic squares were constructed for all listed magic constants less than or equal to 11212.

 Here are the confirmed magic squares:

S = 2862

23 179 409 373 263 137 461 457 523 37

193 353 443 199 317 109 337 397 131 383

71 73 389 251 593 167 439 449 233 197

571 293 101 229 29 557 271 31 379 401

127 419 283 241 269 239 547 89 181 467

491 433 223 113 41 577 43 311 563 67

281 97 163 587 191 313 149 509 421 151

307 499 227 431 103 83 59 479 211 463

277 359 257 331 569 541 53 79 47 349

521 157 367 107 487 139 503 61 173 347

 

S = 3092

41 491 599 487 373 229 541 73 79 179

397 101 137 167 461 127 557 523 263 359

449 251 383 107 197 149 191 521 401 443

569 139 587 479 83 317 181 241 257 239

601 367 109 89 509 157 43 593 277 347

193 463 467 389 281 607 113 97 379 103

163 547 409 499 59 439 223 173 311 269

71 53 61 211 571 563 433 131 577 421

271 613 293 233 227 353 307 283 199 313

337 67 47 431 331 151 503 457 349 419

 

S=3500

71 211 257 223 587 643 443 313 613 139

379 653 491 293 167 227 503 97 439 251

191 563 137 409 569 269 353 523 113 373

521 101 271 617 431 367 73 557 173 389

383 131 311 179 401 359 397 547 283 509

157 499 577 337 233 541 83 347 619 107

421 109 229 197 599 151 419 103 641 631

263 601 457 479 307 239 281 331 193 349

467 433 163 317 79 241 487 593 149 571

647 199 607 449 127 463 461 89 277 181

 

S = 4222

131 149 443 607 647 619 521 499 257 349

139 431 547 293 137 587 523 389 467 709

701 317 359 379 263 577 197 227 571 631

617 509 653 251 673 503 421 151 277 167

191 593 229 449 179 661 397 719 457 347

223 419 269 641 401 601 563 233 463 409

599 479 383 271 613 173 541 461 491 211

557 311 367 659 193 181 199 733 283 739

337 331 281 433 677 157 487 569 643 307

727 683 691 239 439 163 373 241 313 353

 

S = 4780

179 227 617 479 571 599 541 463 331 773

491 311 523 661 251 487 313 691 433 619

439 653 263 701 719 397 751 353 211 293

709 449 257 277 521 683 613 223 587 461

769 641 421 181 733 419 349 431 457 379

271 347 743 337 563 673 191 199 809 647

569 797 317 283 383 241 643 557 601 389

443 757 307 727 409 269 281 787 607 193

233 359 739 373 401 503 467 499 547 659

677 239 593 761 229 509 631 577 197 367

 

S=5608

251 281 809 491 661 619 263 631 863 739

409 857 571 641 593 599 479 389 283 787

659 587 557 577 547 683 827 317 541 313

353 601 347 821 257 769 859 743 509 349

761 431 271 269 331 653 727 773 569 823

379 677 643 673 487 383 719 523 373 751

883 521 881 359 421 563 367 401 709 503

797 439 757 449 647 293 467 733 607 419

839 877 311 499 811 433 443 397 691 307

277 337 461 829 853 613 457 701 463 617

 

S=7124

389 457 853 751 857 809 709 811 719 769

431 773 1013 733 877 971 739 401 677 509

563 823 467 409 421 997 547 887 977 1033

1009 859 587 1021 499 397 617 967 521 647

443 577 727 827 1039 937 593 821 487 673

991 659 439 449 613 881 541 941 691 919

1031 743 619 641 571 503 911 479 1019 607

983 757 829 1049 701 599 797 419 433 557

653 523 929 601 863 569 787 491 761 947

631 953 661 643 683 461 883 907 839 463

 

S=10126

661 829 683 1013 907 1171 1181 1217 1187 1277

1291 1237 809 1279 1063 797 743 1201 883 823

1039 733 857 1307 1097 827 1259 937 709 1361

821 859 1327 953 971 1297 769 1129 1249 751

1049 1019 1321 991 1109 1163 967 727 1103 677

1367 1231 1117 739 887 1087 673 853 1021 1151

701 1093 1009 787 947 977 1229 1319 1303 761

997 1153 1193 983 811 839 1373 863 691 1223

911 941 929 773 1051 1091 1213 1123 1061 1033

1289 1031 881 1301 1283 877 719 757 919 1069

 

S=10198

673 853 1009 859 1123 1063 971 1163 1103 1381

1069 1109 677 809 937 997 1213 1373 1187 827

1171 1193 907 1259 757 701 1249 911 743 1307

1327 1021 1097 863 761 1217 1229 881 709 1093

1087 821 1223 1291 1361 953 787 887 769 1019

857 811 1297 1129 1049 1301 929 1151 991 683

1367 1303 967 829 983 1031 877 941 1061 839

1051 691 719 1201 1277 1237 739 733 1319 1231

773 1117 1013 1039 797 947 1321 1181 1283 727

823 1279 1289 919 1153 751 883 977 1033 1091

 

S = 11212

769 863 1171 967 859 1381 1237 1459 1289 1217

1163 953 797 1297 1049 1021 1303 977 1423 1229

809 1277 1153 937 1151 1409 1291 839 1249 1097

1429 1231 1193 1451 1061 829 821 1361 823 1013

1453 997 947 1091 1321 887 1283 941 811 1481

1069 1201 1427 1129 907 919 1373 1039 1117 1031

1009 1123 1301 1093 1367 1483 911 1051 1087 787

991 1109 1279 877 1223 929 1187 1433 1327 857

1213 1439 1063 971 1447 883 773 1259 983 1181

1307 1019 881 1399 827 1471 1033 853 1103 1319

In the case of 27180, the primes to be organised into the 10 x 10 magic square are:

2339, 2341, 2347, 2351, 2357, 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121