A343194 a(n) is the parameter b in the three-parameter description of 3 X 3 magic squares of consecutive primes (see comment).

12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 30, 12, 30, 12, 12, 12, 30, 12, 12, 30, 12, 12, 30, 12, 30, 12, 18, 12, 12, 30, 12, 30, 12, 18, 12, 12, 12, 12, 30, 12, 12, 60, 30, 12, 12, 12, 30, 30, 12, 6, 30, 30, 18, 18, 42, 12, 12, 42, 12, 12, 18, 12, 12, 12, 12, 30

Offset: 1

Author

A.H.M. Smeets, Apr 07 2021

Keywords

• nonn

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 c is given in A343195.

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), A343195 (c).

Formula

a(n) = (A270305(n) - 3*A256891(n) - 3*A343195(n))/9.

a(n) = (A166113(n) - A256891(n) - A343195(n))/3.