A227909 - OEIS

%I #37 Jul 26 2023 21:04:39

%S 0,1,1,1,2,1,2,3,3,1,2,5,2,3,2,3,3,5,3,1,5,4,5,4,3,4,7,4,4,2,1,4,9,2,

%T 4,11,4,2,6,2,6,11,6,4,3,3,5,6,4,3,6,2,4,10,3,10,12,7,1,6,6,5,11,4,5,

%U 6,4,3,11,2,10,13,4,6,5,2,14,13,2,2,5,5,9,15,5,3,7,8,5,3,5,7,15,3,1,8,5,7,11,4

%N Number of ways to write 2*n = p + q with p, q and (p-1)*(q+1) - 1 all prime.

%C Conjecture: a(n) > 0 for all n > 1.

%C This is stronger than Goldbach's conjecture for even numbers. It also implies A. Murthy's conjecture (cf. A109909) for even numbers.

%C We have verified the conjecture for n up to 2*10^7.

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

%H Zhi-Wei Sun, <a href="/A227909/b227909.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>, preprint, arXiv:1211.1588 [math.NT], 2012-2017.

%e a(6) = 1 since 2*6 = 5 + 7, and (5-1)*(7+1)-1 = 31 is prime.

%e a(10) = 1 since 2*10 = 7 + 13, and (7-1)*(13+1)-1 = 83 is prime.

%e a(20) = 1 since 2*20 = 17 + 23, and (17-1)*(23+1)-1 = 383 is prime.

%t a[n_]:=Sum[If[PrimeQ[2n-Prime[i]]&&PrimeQ[(Prime[i]-1)(2n-Prime[i]+1)-1],1,0],{i,1,PrimePi[2n-2]}]

%t Table[a[n],{n,1,100}]

%Y Cf. A002375, A109909, A218754, A219052, A220431, A220455, A220554, A227908, A230230.

%K nonn

%O 1,5

%A _Olivier Gérard_ and _Zhi-Wei Sun_, Oct 13 2013