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%I #57 Feb 16 2025 08:33:25
%S 1480028129,1850590057,5196185947,5601567187,5757284497,6048371029,
%T 6151077269,9517122259,19052235847,20477868319,23813359613,
%U 24026890159,26748150199,28519991387,34821326119,44420969909,49285771679,73827799009,73974781889,74220519319,76483907837,76560277009,80143089599,85892025227,89132925737,95515449037,99977424653
%N Smallest primes of 3 X 3 magic squares formed from consecutive primes.
%C Let a = a(n) for some n and {a, b, c, d, e, f, g, h, i} be the set of consecutive primes. Then it is:
%C +---+---+---+ +---+---+---+
%C | d | c | h | | c | d | h |
%C +---+---+---+ +---+---+---+
%C | i | e | a | (type 1), or | i | e | a | (type 2). See Harvey D. Heinz.
%C +---+---+---+ +---+---+---+
%C | b | g | f | | b | f | g |
%C +---+---+---+ +---+---+---+
%C The type is determined by the sign of A343195.
%C For a given magic sum S, it is easy to calculate the unique set of n^2 consecutive primes that sum up to n*S (see PROGRAM MagicPrimes() in A073519), and in particular the smallest of these (cf. PROGRAM), listed here for n = 3, in A260673 for n = 4, in A272386 for n = 5, and in A272387 for n = 6. - _M. F. Hasler_, Oct 28 2018
%D Allan W. Johnson, Jr., Consecutive-Prime Magic Squares, Journal of Recreational Mathematics, vol. 15, 1982-83, pp. 17-18.
%D H. L. Nelson, A Consecutive Prime 3 x 3 Magic Square, Journal of Recreational Mathematics, vol. 20:3, 1988, p. 214.
%H A.H.M. Smeets, <a href="/A256891/b256891.txt">Table of n, a(n) for n = 1..759</a>
%H Harvey D. Heinz, <a href="http://www.magic-squares.net/primesqr.htm#Minimum consecutive primes -3">Prime Numbers Magic Squares: Minimum consecutive primes - 3</a>, 1999-2010.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeMagicSquare.html">Prime Magic Square</a>
%H <a href="/index/Mag#magic">Index entries for sequences related to magic squares</a>
%F a(n) = A151799(A151799(A151799(A151799(A166113(n))))). - _Max Alekseyev_, Nov 02 2015
%o (Magma) /* Brute-force search */ lst:=[]; n:=3; while n lt 10^11 do a:=NextPrime(n); q:=a; j:=a-n; if j mod 6 eq 0 then b:=NextPrime(a); if j eq b-a then c:=NextPrime(b); d:=c-b; if d mod 6 eq 0 then e:=NextPrime(c); k:=e-c; if k eq j then f:=NextPrime(e); if k eq f-e then g:=NextPrime(f); if g-f eq d then h:=NextPrime(g); m:=h-g; if m eq k then i:=NextPrime(h); if h-g eq i-h then Append(~lst, n); end if; end if; end if; end if; end if; end if; end if; end if; n:=q; end while; lst;
%o (PARI) A256891(n)=MagicPrimes(A270305(n),3)[1] \\ See A073519 for MagicPrimes(). - _M. F. Hasler_, Oct 28 2018
%Y Cf. A073519, A151799, A166113, A260673, A272386, A272387, A343194, A343195.
%Y Subsequence of A265139.
%K nonn,changed
%O 1,1
%A _Arkadiusz Wesolowski_, Apr 12 2015
%E Extended by _Max Alekseyev_, Nov 02 2015