A377014 a(n) is the number of primes p such that p - 6, p + 6 and 2*n - p are also primes.

0, 0, 0, 0, 0, 0, 1, 2, 2, 2, 2, 3, 2, 3, 4, 1, 3, 3, 0, 4, 4, 2, 2, 3, 3, 3, 6, 3, 4, 6, 0, 5, 5, 1, 6, 4, 3, 5, 6, 4, 3, 9, 3, 2, 8, 2, 4, 7, 2, 4, 3, 3, 5, 5, 6, 4, 9, 4, 4, 11, 2, 5, 10, 1, 4, 4, 4, 4, 4, 5, 2, 7, 4, 4, 9, 2, 5, 6, 0, 6, 7, 5, 3, 6, 5, 1, 10, 7, 4, 9, 2, 5, 9, 2, 6, 5, 4, 5, 4, 4

Offset: 1

Author

Lei Zhou, Oct 12 2024

Keywords

• nonn
• easy

Comments

Conjecture: a(n) = 0 only when n = 1, 2, 3, 4, 5, 6, 19, 31, 331, 499.

Examples

			a(7) = 1 since only when p = 11 are p - 6, p + 6 and 2n - p all prime.
a(12) = 3 from the cases when p is 11, 13 or 17:
  when p = 11, {p - 6, p + 6, 2n - p} = {5, 17, 13} are all prime;
  when p = 13, {p - 6, p + 6, 2n - p} = {7, 13, 19, 11} are all prime;
  when p = 17, {p - 6, p + 6, 2n - p} = {11, 17, 23, 7} are all prime.
a(19) = 0 since 2n = 38 = 7 + 31 = 19 + 19 = 31 + 7, and none of p = 7, 19, 31 can make p - 6 and p + 6 both prime.
		

Crossrefs

Cf. A006489, A045917.

Programs

Maple

f:= proc(n) local i;
  nops(select(p -> andmap(isprime,[p,p-6,p+6, 2*n-p]), [seq(i,i=3..2*n,2)]))
end proc:
map(f, [$1..100]); # Robert Israel, Oct 13 2024

Mathematica

m = 200; ps = {}; p = 7; While[p = NextPrime[p]; If[PrimeQ[p - 6] && PrimeQ[p + 6], AppendTo[ps, p]]; p < 2*m]; a = {}; Do[ct = 0; k = 0; While[k++; ps[[k]] < n, q = n - ps[[k]]; If[PrimeQ[q], ct++]]; AppendTo[a, ct]; If[ct == 0, AppendTo[b, n]], {n, 2, m, 2}]; a