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

1,5

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.

Links

Table of n, a(n) for n=1..84.

Example

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.

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]

Crossrefs

Cf. A240712, A171611, A254041.

Sequence in context: A271319 A367404 A319178 * A254041 A240712 A171611

Adjacent sequences: A376507 A376508 A376509 * A376511 A376512 A376513

Keyword

nonn,easy

Author

Lei Zhou, Sep 25 2024

Status

approved