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%I #8 Jun 20 2020 16:23:32
%S 0,3,5,7,7,11,13,13,17,19,19,23,23,19,29,31,31,0,37,37,41,43,43,47,47,
%T 43,53,53,43,59,61,61,0,67,67,71,73,73,0,79,79,83,83,79,89,89,79,0,97,
%U 97,101,103,103,107,109,109,113,113,109,0,113,109,0,127,127,131,131,127,137,139,139
%N Maximal common prime of two Goldbach partitions of 2n and 2(n+1) or zero (if common prime does not exist).
%H <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%e 4 = 2+2 and 6 = 3+3. Since those are the only available Goldbach partitions and they have no common prime, a(4/2) = a(2) = 0. 14 = 3+11 and 16 = 5+11, so a(14/2) = a(7) = 11.
%p S:= proc(n) option remember; {seq((h-> `if`(
%p andmap(isprime, h), h, [])[])([n+i, n-i]), i=0..n-2)}
%p end:
%p a:= n-> max(0, (S(n) intersect S(n+1))[]):
%p seq(a(n), n=2..80); # _Alois P. Heinz_, Jun 20 2020
%t d[n_]:=Flatten[Cases[FrobeniusSolve[{1,1},2*n],{__?PrimeQ}]]
%t e[n_]:=Intersection[d[n],d[n+1]]; f[n_]:=If[e[n]=={},0,Max[e[n]]];
%t f/@Range[2,100]
%Y Cf. A002372, A002373, A002375, A045917, A060308, A335045.
%K nonn
%O 2,2
%A _Ivan N. Ianakiev_, May 21 2020
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