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.

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

Offset: 1

Author

Lei Zhou, Sep 25 2024

Keywords

• nonn
• easy

Comments

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

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.

Examples

			For n=1, 2*(n+4)=10, 10=5+5, and 5+6=11 is a prime. Thus a(1)=1;
For n=2, 2*(n+4)=12, 12=5+7, and 5+6=11 is a prime. Thus a(2)=1;
...
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.
		

Crossrefs

Cf. A240712, A171611, A254041.

Programs

Mathematica

res = {}; Do[n[2] = i*6; n[1] = n[2] - 2; n[3] = n[2] + 2;
 Do[c[j] = 0; p[j] = NextPrime[n[j]/2 - 1];
  While[q[j] = n[j] - p[j];
   If[PrimeQ[q[j]] && q[j] > 3,
    If[PrimeQ[p[j] + 6] || PrimeQ[q[j] + 6], c[j]++]];
   p[j] < n[j] - 5, p[j] = NextPrime[p[j]]], {j, 1, 3}];
 AppendTo[res, c[1]]; AppendTo[res, c[2]];
 AppendTo[res, c[3]], {i, 2, 29}]; Print[res]