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A376510 a(n) is the number of pairs of primes p+q=2*(n+4) with 5 <= p <= n such that either p+6 or q+6 is also prime.

%I A376510 #17 Oct 21 2024 14:30:07
%S A376510 1,1,1,1,2,1,2,3,2,2,3,1,3,4,1,2,4,2,3,4,3,3,5,2,3,6,1,4,6,2,4,5,4,4,
%T A376510 6,4,4,8,3,3,8,3,5,7,2,4,7,4,5,6,5,6,9,5,4,12,3,5,10,2,5,7,5,5,6,6,5,
%U A376510 11,5,4,11,2,7,8,3,6,10,5,4,9,7,5,11,6
%N A376510 a(n) is the number of pairs of primes p+q=2*(n+4) with 5 <= p <= n such that either p+6 or q+6 is also prime.
%C A376510 It is hypothesized that all terms of this sequence are positive integers.
%C A376510 If the above hypothesis is true, the Goldbach Hypothesis is true, since for every even number 2n, if there is a Goldbach decomposition p+q=2n meets the condition of this sequence, p+q+6=2n+6 forms at least one Goldbach decomposition of 2n+6.
%e A376510 For n=1, 2*(n+4)=10, 10=5+5, and 5+6=11 is a prime. Thus a(1)=1;
%e A376510 For n=2, 2*(n+4)=12, 12=5+7, and 5+6=11 is a prime. Thus a(2)=1;
%e A376510 ...
%e A376510 For n=14, 2*(n+4)=36, 36=5+31 (5+6=11); 7+29 (7+6=13); 13+23 (13+6=19); 17+19 (17+6=23), four cases found.  Thus a(14)=4.
%t A376510 res = {}; Do[n[2] = i*6; n[1] = n[2] - 2; n[3] = n[2] + 2;
%t A376510  Do[c[j] = 0; p[j] = NextPrime[n[j]/2 - 1];
%t A376510   While[q[j] = n[j] - p[j];
%t A376510    If[PrimeQ[q[j]] && q[j] > 3,
%t A376510     If[PrimeQ[p[j] + 6] || PrimeQ[q[j] + 6], c[j]++]];
%t A376510    p[j] < n[j] - 5, p[j] = NextPrime[p[j]]], {j, 1, 3}];
%t A376510  AppendTo[res, c[1]]; AppendTo[res, c[2]];
%t A376510  AppendTo[res, c[3]], {i, 2, 29}]; Print[res]
%Y A376510 Cf. A240712, A171611, A254041.
%K A376510 nonn,easy
%O A376510 1,5
%A A376510 _Lei Zhou_, Sep 25 2024