cp's OEIS Frontend

A220091 Number of ways to write n=p+q+(n mod 2)q with p>q and p, q, 6q-1, 6q+1 all prime.

%I A220091 #8 Jul 23 2025 00:52:53
%S A220091 0,0,0,0,0,0,1,1,1,1,2,1,1,1,1,2,3,2,1,2,2,2,3,2,2,2,3,1,2,1,1,1,3,2,
%T A220091 2,3,2,1,1,2,2,1,2,2,2,3,3,3,1,2,3,2,3,3,1,1,4,2,1,3,1,1,3,4,3,3,2,1,
%U A220091 1,3,3,1,2,2,4,4,5,3,1,1,3,2,3,3,2,2,4,2,3,3,0,1,5,2,2,3,1,0,2,3
%N A220091 Number of ways to write n=p+q+(n mod 2)q with p>q and p, q, 6q-1, 6q+1 all prime.
%C A220091 Conjecture: a(n)>0 for all even n>=8070 and odd n>=18680.
%C A220091 This conjecture unifies the twin prime conjecture, Goldbach's conjecture and Lemoine's conjecture. It has been verified for n up to 10^7.
%C A220091 Zhi-Wei Sun also made the following conjecture: Any integer n>=6782 can be written as p+q+(n mod 2)q with p>q and p, q, q-6, q+6 all prime, and any integer n>=4410 can be written as p+q+(n mod 2)q with p>q and p, q, 2q-3, 2q+3 all prime, and any integer n>=16140 can be written as p+q+(n mod 2)q with p>q and p, q, 3q-2, 3q+2 all prime.
%H A220091 Zhi-Wei Sun, <a href="/A220091/b220091.txt">Table of n, a(n) for n = 1..100000</a>
%H A220091 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588.
%e A220091 a(31)=1 since 31=17+2*7 with 6*7-1 and 6*7+1 twin primes.
%e A220091 a(32)=1 since 32=29+3 with 6*3-1 and 6*3+1 twin primes.
%t A220091 a[n_]:=a[n]=Sum[If[PrimeQ[6Prime[k]-1]==True&&PrimeQ[6Prime[k]+1]==True&&PrimeQ[n-(1+Mod[n,2])Prime[k]]==True,1,0],{k,1,PrimePi[(n-1)/(2+Mod[n,2])]}]
%t A220091 Do[Print[n," ",a[n]],{n,1,100}]
%Y A220091 Cf. A001359, A006512, A002375, A046927, A219185, A219157, A218867, A219055, A219966.
%K A220091 nonn
%O A220091 1,11
%A A220091 _Zhi-Wei Sun_, Dec 04 2012