cp's OEIS Frontend

A187758 Number of ways to write n=x+y (x,y>0) with 2x-3, 2x+3, 6y+1 and 6y+5 all prime.

%I A187758 #11 Jul 22 2025 10:55:03
%S A187758 0,0,0,0,1,2,2,2,2,3,4,2,2,3,3,3,2,3,3,4,5,3,6,5,4,6,3,5,4,3,6,2,4,5,
%T A187758 5,4,4,6,5,4,6,5,4,5,7,5,2,3,6,4,5,4,5,7,6,9,5,4,9,5,4,5,5,4,5,6,3,8,
%U A187758 5,8,8,3,7,5,3,5,3,5,4,9,6,4,9,7,5,8,7,8,6,9,8,2,7,7,5,6,2,10,6,3
%N A187758 Number of ways to write n=x+y (x,y>0) with 2x-3, 2x+3, 6y+1 and 6y+5 all prime.
%C A187758 Conjecture: a(n)>0 for all n>4.
%C A187758 This has been verified for n up to 10^8. It implies that there are infinitely many cousin primes and also infinitely many sexy primes.
%H A187758 Zhi-Wei Sun, <a href="/A187758/b187758.txt">Table of n, a(n) for n = 1..20000</a>
%H A187758 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588.
%e A187758 a(5)=1 since 5=4+1 with 2*4-3, 2*4+3, 6*1+1 and 6*1+5 all prime.
%t A187758 a[n_]:=a[n]=Sum[If[PrimeQ[2k-3]==True&&PrimeQ[2k+3]==True&&PrimeQ[6(n-k)+1]==True&&PrimeQ[6(n-k)+5]==True,1,0],{k,1,n-1}]
%t A187758 Do[Print[n," ",a[n]],{n,1,100}]
%Y A187758 Cf. A023200, A046132, A023201, A002375, A187757, A199920, A219055, A218867, A220455.
%K A187758 nonn
%O A187758 1,6
%A A187758 _Zhi-Wei Sun_, Jan 03 2013