A343195 a(n) is the parameter c in the three parameter description of 3 X 3 magic squares of consecutive primes (see comment).

6, 6, 6, 6, 6, 6, 6, 6, 6, 6, -6, 6, 24, 6, 6, 6, -18, 6, 6, -18, 6, 6, -18, 6, -18, 6, 24, 6, 6, -6, 6, -18, 6, 24, 6, 6, 6, 6, -18, 6, 6, -54, -18, 6, 6, 6, -18, 12, 6, 78, 12, -18, 24, 24, -24, 6, 6, 6, 6, 6, 24, 6, 6, 6, 6, 12, 12, 24, 6, 6, 24, -18, 6, 24

Offset: 1

Author

A.H.M. Smeets, Apr 07 2021

Keywords

• sign

Comments

Each 3 X 3 magic square of consecutive primes can be described by three parameters: p1, b and c, where p1 is the smallest prime in the magic square, b > 0 and c > -b; the magic square is then given by:

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

| p1+5b+2c | p1 | p1+4b+c |

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

| p1+2b | p1+3b+c | p1+4b+2c |

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

| p1+2b+c | p1+6b+2c | p1+b |

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

p1 is given in A256891 and b is given in A343194.

If c > 0, the magic square is of type 1; if -b < c < 0, the magic square is of type 2. If the consecutive primes are given by p1, p2, ..., p9 (in increasing order), the magic square types are given by:

Type 1 Type 2

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

| p8 | p1 | p6 | | p8 | p1 | p7 |

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

| p3 | p5 | p7 | | p4 | p5 | p6 |

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

| p4 | p9 | p2 | | p3 | p9 | p2 |

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

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.
• A.H.M. Smeets, Python program
• Eric Weisstein's World of Mathematics, Prime Magic Square
• Index entries for sequences related to magic squares

Crossrefs

Cf. A166113 (p5), A256891 (p1), A270305 (magic constant), A343194 (b).

Formula

a(n) = (A270305(n) - 3*A256891(n) - 9*A343194(n))/3.

a(n) = A166113(n) - A256891(n) - 3*A343194(n).