cp's OEIS Frontend

A219025 Number of primes p such that 6n-p and 6n+p are both prime.

%I A219025 #14 Jul 23 2025 00:10:58
%S A219025 0,0,0,0,0,1,1,1,1,1,2,1,2,2,2,2,1,1,2,4,1,2,1,3,2,2,2,2,3,2,3,1,1,2,
%T A219025 5,2,2,2,4,3,3,4,1,2,5,3,2,2,5,4,1,3,1,3,5,3,3,3,3,4,4,2,6,4,7,5,2,3,
%U A219025 3,7,5,3,5,5,7,4,4,2,3,4,2,3,3,6,6,3,2,5,4,7,3,4,2,3,7,1,6,4,5,6
%N A219025 Number of primes p<n such that 6n-p and 6n+p are both prime.
%C A219025 Conjecture: a(n)>0 for all n=6,7,...
%C A219025 This has been verified for n up to 10^8.
%C A219025 Zhi-Wei Sun also made the following general conjecture:
%C A219025 Let P(x) be any non-constant integer-valued polynomial with positive leading coefficient. If n is large enough, then there is a prime p<n such that 6P(n)+p and 6P(n)-p are both prime. For example, for P(x)=x(x+1)/2, x^2, x^3, x^4 it suffices to require that n is greater than 1933, 2426, 6772, 24979 respectively.
%C A219025 See also A219023 for similar conjectures.
%H A219025 Zhi-Wei Sun, <a href="/A219025/b219025.txt">Table of n, a(n) for n = 1..20000</a>
%H A219025 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv preprint arXiv:1211.1588, 2012.
%e A219025 a(11)=2 since the 5 and 7 are the only primes p<11 with 66-p and 66+p both prime.
%t A219025 a[n_]:=a[n]=Sum[If[PrimeQ[6n-Prime[k]]==True&&PrimeQ[6n+Prime[k]]==True,1,0],{k,1,PrimePi[n-1]}]
%t A219025 Do[Print[n," ",a[n]],{n,1,20000}]
%Y A219025 Cf. A000040, A002375, A218754, A218585, A218654, A218656, A218825.
%K A219025 nonn
%O A219025 1,11
%A A219025 _Zhi-Wei Sun_, Nov 10 2012