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A256891
Smallest primes of 3 X 3 magic squares formed from consecutive primes.
12
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.
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
KEYWORD
nonn
AUTHOR
EXTENSIONS
Extended by Max Alekseyev, Nov 02 2015
STATUS
approved