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A268790 Magic sums of 3 X 3 magic squares composed of primes.

%I A268790 #40 Apr 03 2023 10:36:13
%S A268790 177,213,219,267,309,327,381,393,411,417,447,453,471,501,519,537,573,
%T A268790 579,633,681,717,723,753,771,789,807,813,843,849,879,921,933,1011,
%U A268790 1041,1047,1059,1077,1101,1119,1137,1149,1167,1191,1203,1227,1257,1263,1293
%N A268790 Magic sums of 3 X 3 magic squares composed of primes.
%C A268790 From _Robert Israel_, Feb 16 2016: (Start)
%C A268790 All terms are 3 times odd primes.
%C A268790 3*p is a term if and only if p is a prime not in A073350.
%C A268790 Conjecture: 3*p is a term for every prime > 859.
%C A268790 I verified this for all primes < 100000.
%C A268790 The Green-Tao theorem implies the sequence is infinite: given one magic square with entries a(i,j), there are infinitely many pairs of positive integers x,y such that b(i,j) = x + y*a(i,j) are all prime.  Then b(i,j) form another magic square. (End)
%C A268790 Every number of the form 3*(A227284(n) + 840) is in this sequence. - _Arkadiusz Wesolowski_, Feb 22 2016
%C A268790 The terms equal three times the central elements (and equivalently, one third of the sum of all elements) of the 3 X 3 magic squares made of primes, which are listed in A320872. - _M. F. Hasler_, Oct 28 2018
%H A268790 Robert Israel, <a href="/A268790/b268790.txt">Table of n, a(n) for n = 1..9552</a>
%H A268790 G. L. Honaker, Jr. and Chris Caldwell, <a href="https://t5k.org/curios/cpage/1341.html">Prime Curios!: 859</a>
%H A268790 Wikipedia, <a href="https://en.wikipedia.org/wiki/Magic_square">Magic square</a>
%H A268790 <a href="/index/Mag#magic">Index entries for sequences related to magic squares</a>
%F A268790 If conjecture is true, a(n) = 3*prime(n+40) for n >= 110. - _Robert Israel_, Feb 16 2016
%F A268790 A268790 = 3*{column 5 of A320872} as a set, i.e., with duplicates removed. - _M. F. Hasler_, Oct 28 2018
%e A268790 Examples of 3 X 3 magic squares composed of primes.
%e A268790 .
%e A268790 +---+---+---+
%e A268790 | 17| 89| 71|
%e A268790 +---+---+---+
%e A268790 |113| 59| 5 |
%e A268790 +---+---+---+
%e A268790 | 47| 29|101|
%e A268790 +---+---+---+
%e A268790 The magic constant is 177 = a(1).
%e A268790 .
%e A268790 +---+---+---+
%e A268790 | 41| 89| 83|
%e A268790 +---+---+---+
%e A268790 |113| 71| 29|
%e A268790 +---+---+---+
%e A268790 | 59| 53|101|
%e A268790 +---+---+---+
%e A268790 The magic constant is 213 = a(2).
%p A268790 N:= 10000: # to get all terms <= N P:= select(isprime,{seq(p,p=3..2*N/3,2)}):
%p A268790 count:= 0:
%p A268790 for ic from 1 while P[ic] <= N/3 do
%p A268790    c:= P[ic];
%p A268790    V:= map(`-`,P[ic+1..-1],c) intersect map(t -> c-t, P[1..ic-1]);
%p A268790    nv:= nops(V);
%p A268790    VV:= {seq(seq(V[j]-V[i],j=i+1..nv),i=1..nv-1)} intersect V;
%p A268790    nvv:= nops(VV);
%p A268790    found:= false;
%p A268790    for ia from 1 to nvv while not found do
%p A268790      a:= VV[ia];
%p A268790      for ib from ia+1 to nvv while VV[ib] < c - a do
%p A268790        b:= VV[ib];
%p A268790        if b <> 2*a and {c-a-b,c-a+b,c-b+a,c+a+b} subset P then
%p A268790           found:= true;
%p A268790           count:= count+1;
%p A268790           A[count]:= 3*c;
%p A268790           break
%p A268790        fi
%p A268790      od
%p A268790    od:
%p A268790 od:
%p A268790 seq(A[i],i=1..count); # _Robert Israel_, Feb 16 2016
%o A268790 (PARI) c=3;A268790_vec=3*vector(50,i,c=A320872_row(1,0,c+1)[2,2]) \\ Illustrates formula & comment. - _M. F. Hasler_, Oct 28 2018
%o A268790 (PARI) is_A268790(c)={denominator(c/=3)==1&& isprime(c)&& forstep(a=2,c\2-1,2, isprime(c-a)&& isprime(c+a)&& forstep(b=2,c-2*a-2,2, isprime(c-2*a-b)&& isprime(c-a-b)&& isprime(c-b)&& isprime(c+b)&& isprime(c+a+b)&& isprime(c+2*a+b)&& return(1)))} \\ _M. F. Hasler_, Oct 28 2018
%Y A268790 Cf. A000040, A024351, A073350, A164843, A227284.
%Y A268790 Cf. A320872, A320871, A320873.
%K A268790 nonn
%O A268790 1,1
%A A268790 _Arkadiusz Wesolowski_, Feb 13 2016