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%I #28 Oct 28 2019 04:20:51
%S 2211,2261,2311,2463,2725,4257,6125,6611,7821,9841,9973,10303,10499,
%T 10631,10953,11987,12115,12179,12243,12309,12375,12637,12837,13497,
%U 13695,14169,15063,15395,16207,16483,16821,17605,17891,19017,20345,20487,21135,22539,22811,23219,23985
%N Potential magic constants of 9 X 9 magic squares composed of consecutive primes.
%C For a 9 X 9 magic square composed of 81 consecutive primes, it is necessary that the sum of these primes is a multiple of 9.
%C This sequence consists of integers equal the sum of 81 consecutive primes divides by 9. It is not known whether each such set of consecutive primes can be arranged into 9 X 9 magic square but it looks plausible.
%H Stefano Tognon, <a href="http://digilander.libero.it/ice00/magic/prime/squares37.html#9">Squares from 37</a> (in Italian).
%H Natalia Makarova, <a href="http://www.natalimak1.narod.ru/prime9.htm">Sequence of Magic Numbers MK 9th Order</a> (in Russian).
%e a(1)=2211 for a square containing prime(12)..prime(92):
%e [37 127 163 179 229 233 379 421 443
%e 41 431 463 457 59 139 433 109 79
%e 409 311 389 71 307 347 281 53 43
%e 373 137 181 251 401 239 317 89 223
%e 173 419 101 103 113 353 313 277 359
%e 97 383 397 479 47 197 107 263 241
%e 349 131 193 149 367 199 73 467 283
%e 439 61 257 191 227 167 151 449 269
%e 293 211 67 331 461 337 157 83 271]
%e a(2)=2261 for a square containing prime(13)..prime(93):
%e [41 379 281 467 349 257 229 199 59
%e 313 223 127 337 131 101 479 107 443
%e 409 71 331 79 137 263 347 271 353
%e 211 307 487 149 251 293 181 113 269
%e 191 419 109 439 173 233 103 397 197
%e 97 283 193 317 433 457 241 157 83
%e 461 139 239 359 373 179 67 401 43
%e 89 277 73 53 367 167 463 389 383
%e 449 163 421 61 47 311 151 227 431]
%p s:= proc(n) option remember;
%p `if` (n=1, add (ithprime(i), i=1..81),
%p ithprime(n+80) -ithprime(n-1) +s(n-1))
%p end:
%p a:= proc(n) option remember; local k, m;
%p a(n-1);
%p for k from 1+b(n-1) while irem (s(k), 9, 'm')<>0 do od;
%p b(n):= k; m
%p end:
%p a(0):=0: b(0):=0:
%p seq (a(n), n=1..50);
%t Total[#]/9&/@Select[Partition[Prime[Range[500]],81,1],Divisible[ Total[ #],9]&] (* _Harvey P. Dale_, Jan 08 2014 *)
%Y Cf. A073520, A173981, A176571, A177434, A188536, A189188.
%K nonn
%O 1,1
%A _Natalia Makarova_, Jun 11 2011
%E Edited by _Max Alekseyev_, Jun 18 2011
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