A256891 Smallest primes of 3 X 3 magic squares formed from consecutive primes.

1480028129, 1850590057, 5196185947, 5601567187, 5757284497, 6048371029, 6151077269, 9517122259, 19052235847, 20477868319, 23813359613, 24026890159, 26748150199, 28519991387, 34821326119, 44420969909, 49285771679, 73827799009, 73974781889, 74220519319, 76483907837, 76560277009, 80143089599, 85892025227, 89132925737, 95515449037, 99977424653

Offset

1,1

Comments

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:

+---+---+---+ +---+---+---+

| d | c | h | | c | d | h |

+---+---+---+ +---+---+---+

| i | e | a | (type 1), or | i | e | a | (type 2). See Harvey D. Heinz.

+---+---+---+ +---+---+---+

| b | g | f | | b | f | g |

+---+---+---+ +---+---+---+

The type is determined by the sign of A343195.

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

References

Allan W. Johnson, Jr., Consecutive-Prime Magic Squares, Journal of Recreational Mathematics, vol. 15, 1982-83, pp. 17-18.

H. L. Nelson, A Consecutive Prime 3 x 3 Magic Square, Journal of Recreational Mathematics, vol. 20:3, 1988, p. 214.

Links

A.H.M. Smeets, Table of n, a(n) for n = 1..759

Harvey D. Heinz, Prime Numbers Magic Squares: Minimum consecutive primes - 3, 1999-2010.

Eric Weisstein's World of Mathematics, Prime Magic Square

Index entries for sequences related to magic squares

Formula

a(n) = A151799(A151799(A151799(A151799(A166113(n))))). - Max Alekseyev, Nov 02 2015

Prog

(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;
(PARI) A256891(n)=MagicPrimes(A270305(n), 3)[1] \\ See A073519 for MagicPrimes(). - M. F. Hasler, Oct 28 2018

Crossrefs

Cf. A073519, A151799, A166113, A260673, A272386, A272387, A343194, A343195.

Subsequence of A265139.

Sequence in context: A048051 A345337 A073519 * A320873 A166113 A157816

Adjacent sequences: A256888 A256889 A256890 * A256892 A256893 A256894

Keyword

nonn

Author

Arkadiusz Wesolowski, Apr 12 2015

Extensions

Extended by Max Alekseyev, Nov 02 2015

Status

approved