A220431 - OEIS

%I #14 Jul 29 2023 03:22:31

%S 0,0,0,1,2,2,2,2,3,2,2,3,2,2,3,3,2,3,2,2,6,1,3,6,4,3,3,2,3,4,3,4,2,3,

%T 3,5,4,4,7,1,2,5,1,5,7,4,2,3,7,4,7,2,4,7,4,4,5,2,5,8,4,3,3,5,2,8,5,4,

%U 3,10,7,8,2,3,5,5,3,6,3,3,14,4,3,12,3,7,7,5,6,8,7,5,9,9,4,4,3,6,10,8

%N Number of ways to write n=x+y (x>0, y>0) with 3x-1, 3x+1 and xy-1 all prime.

%C Conjecture: a(n)>0 for all n>3.

%C This has been verified for n up to 10^8, and it is stronger than A. Murthy's conjecture related to A109909.

%C Conjecture verified for n up to 10^9. - _Mauro Fiorentini_, Jul 26 2023

%C The conjecture implies the twin prime conjecture for the following reason: If x_1<...<x_k are positive integers and q_1,...,q_k are distinct primes greater than x_k, then by the Chinese Remainder Theorem there are infinitely many positive integers n such that x_i(n-x_i) == 1 (mod q_i).

%C Zhi-Wei Sun also made some similar conjectures. For example, any integer n>2 not equal to 63 can be written as x+y (x>0, y>0) with 2x-1, 2x+1 and 2xy+1 all prime.

%C Conjecture verified for n up to 10^9. - _Mauro Fiorentini_, Jul 26 2023

%H Zhi-Wei Sun, <a href="/A220431/b220431.txt">Table of n, a(n) for n = 1..10000</a>

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588.

%e a(22)=1 since 22=4+18 with 3*4-1, 3*4+1 and 4*18-1 all prime.

%t a[n_]:=a[n]=Sum[If[PrimeQ[3k-1]==True&&PrimeQ[3k+1]==True&&PrimeQ[k(n-k)-1]==True,1,0],{k,1,n-1}]

%t Do[Print[n," ",a[n]],{n,1,1000}]

%Y Cf. A001359, A006512, A220419, A220413, A173587, A220272, A219842, A219864, A219923.

%K nonn

%O 1,5

%A _Zhi-Wei Sun_, Dec 14 2012