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

%I A343194 #22 Feb 16 2025 08:34:01
%S A343194 12,12,12,12,12,12,12,12,12,12,30,12,30,12,12,12,30,12,12,30,12,12,30,
%T A343194 12,30,12,18,12,12,30,12,30,12,18,12,12,12,12,30,12,12,60,30,12,12,12,
%U A343194 30,30,12,6,30,30,18,18,42,12,12,42,12,12,18,12,12,12,12,30
%N A343194 a(n) is the parameter b in the three-parameter description of 3 X 3 magic squares of consecutive primes (see comment).
%C A343194 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:
%C A343194   +----------+----------+----------+
%C A343194   | p1+5b+2c | p1       | p1+4b+c  |
%C A343194   +----------+----------+----------+
%C A343194   | p1+2b    | p1+3b+c  | p1+4b+2c |
%C A343194   +----------+----------+----------+
%C A343194   | p1+2b+c  | p1+6b+2c | p1+b     |
%C A343194   +----------+----------+----------+
%C A343194 p1 is given in A256891 and c is given in A343195.
%C 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:
%C A343194   Type 1             Type 2
%C A343194   +----+----+----+   +----+----+----+
%C A343194   | p8 | p1 | p6 |   | p8 | p1 | p7 |
%C A343194   +----+----+----+   +----+----+----+
%C A343194   | p3 | p5 | p7 |   | p4 | p5 | p6 |
%C A343194   +----+----+----+   +----+----+----+
%C A343194   | p4 | p9 | p2 |   | p3 | p9 | p2 |
%C A343194   +----+----+----+   +----+----+----+
%H A343194 A.H.M. Smeets, <a href="/A343194/b343194.txt">Table of n, a(n) for n = 1..759</a>
%H A343194 Harvey D. Heinz, <a href="http://www.magic-squares.net/primesqr.htm#Minimum consecutive primes -3">Prime Numbers Magic Squares: Minimum consecutive primes - 3</a>, 1999-2010.
%H A343194 A.H.M. Smeets, <a href="/A343195/a343195.txt">Python program</a>
%H A343194 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeMagicSquare.html">Prime Magic Square</a>
%H A343194 <a href="/index/Mag#magic">Index entries for sequences related to magic squares</a>
%F A343194 a(n) = (A270305(n) - 3*A256891(n) - 3*A343195(n))/9.
%F A343194 a(n) = (A166113(n) - A256891(n) - A343195(n))/3.
%Y A343194 Cf. A166113 (p5), A256891 (p1), A270305 (magic constant), A343195 (c).
%K A343194 nonn
%O A343194 1,1
%A A343194 _A.H.M. Smeets_, Apr 07 2021