login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

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.
0

%I #17 Oct 21 2024 14:30:07

%S 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 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 11,5,4,11,2,7,8,3,6,10,5,4,9,7,5,11,6

%N 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 It is hypothesized that all terms of this sequence are positive integers.

%C 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 For n=1, 2*(n+4)=10, 10=5+5, and 5+6=11 is a prime. Thus a(1)=1;

%e For n=2, 2*(n+4)=12, 12=5+7, and 5+6=11 is a prime. Thus a(2)=1;

%e ...

%e 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 res = {}; Do[n[2] = i*6; n[1] = n[2] - 2; n[3] = n[2] + 2;

%t Do[c[j] = 0; p[j] = NextPrime[n[j]/2 - 1];

%t While[q[j] = n[j] - p[j];

%t If[PrimeQ[q[j]] && q[j] > 3,

%t If[PrimeQ[p[j] + 6] || PrimeQ[q[j] + 6], c[j]++]];

%t p[j] < n[j] - 5, p[j] = NextPrime[p[j]]], {j, 1, 3}];

%t AppendTo[res, c[1]]; AppendTo[res, c[2]];

%t AppendTo[res, c[3]], {i, 2, 29}]; Print[res]

%Y Cf. A240712, A171611, A254041.

%K nonn,easy

%O 1,5

%A _Lei Zhou_, Sep 25 2024