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
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
Arkadiusz Wesolowski, Apr 12 2015
EXTENSIONS
Extended by Max Alekseyev, Nov 02 2015
STATUS
approved