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

1,1

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

Formula

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

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

Crossrefs

Cf. A166113 (p5), A256891 (p1), A270305 (magic constant), A343195 (c).

Sequence in context: A069873 A129498 A010851 * A123896 A122878 A302341

Adjacent sequences: A343191 A343192 A343193 * A343195 A343196 A343197

Keyword

nonn

Author

A.H.M. Smeets, Apr 07 2021

Status

approved