cp's OEIS Frontend

A219923 Number of ways to write n=x+y (x>0, y>0) with x-1, x+1 and 2*x*y+1 all prime.

%I A219923 #8 Jul 23 2025 00:47:33
%S A219923 0,0,0,0,0,1,1,0,2,0,1,1,1,1,3,2,0,1,2,2,3,2,1,0,2,2,0,1,3,2,2,1,3,4,
%T A219923 2,2,3,0,4,3,3,1,1,3,0,3,2,1,1,3,3,1,1,5,3,1,2,1,3,3,5,3,1,2,4,3,3,2,
%U A219923 4,3,2,2,0,3,5,4,1,3,6,2,6,2,2,4,5,5,2,3,3,4,1,2,0,1,4,2,4,1,6,6
%N A219923 Number of ways to write n=x+y (x>0, y>0) with x-1, x+1 and 2*x*y+1 all prime.
%C A219923 Conjecture: a(n)>0 for all n>623.
%C A219923 This has been verified for n up to 10^8.
%C A219923 Zhi-Wei Sun made the following general conjecture: For each nonnegative integer m, any sufficiently large integer n can be written as x+y (x>0, y>0) with x-m, x+m and 2*x*y+1 all prime.
%C A219923 For example, when m = 2, 3, 4, 5 it suffices to require that n is greater than 28, 151, 357, 199 respectively.
%C A219923 Sun also conjectured that for each m=0,1,2,... any sufficiently large integer n with m or n odd can be written as x+y (x>0, y>0) with x-m, x+m and x*y-1 all prime.
%C A219923 For example, in the case m=1 it suffices to require that n is greater than 4 and not among 40, 125, 155, 180, 470, 1275, 2185, 3875; when m=2 it suffices to require that n is odd, greater than 7, and different from 13.
%H A219923 Zhi-Wei Sun, <a href="/A219923/b219923.txt">Table of n, a(n) for n = 1..10000</a>
%H A219923 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588.
%e A219923 a(11)=1 since 11=6+5 with 6-1, 6+1 and 2*6*5+1=61 all prime.
%t A219923 a[n_]:=a[n]=Sum[If[PrimeQ[Prime[k]+2]==True&&PrimeQ[2(Prime[k]+1)(n-Prime[k]-1)+1]==True,1,0],{k,1,PrimePi[n-1]}]
%t A219923 Do[Print[n," ",a[n]],{n,1,10000}]
%Y A219923 Cf. A219864, A219842.
%K A219923 nonn
%O A219923 1,9
%A A219923 _Zhi-Wei Sun_, Dec 01 2012