A230254 - OEIS

%I #9 Oct 14 2013 03:44:47

%S 0,0,0,1,1,2,1,2,2,2,3,2,1,4,1,2,5,2,3,2,3,4,4,3,4,4,2,2,8,1,6,6,2,3,

%T 2,3,5,5,5,1,5,3,7,5,1,7,10,1,3,4,8,5,3,3,3,5,8,4,10,2,9,3,3,4,7,5,9,

%U 5,4,3,15,4,12,7,4,5,9,3,11,4,6,5,9,5,6,12,6,5,8,1,4,8,5,13,9,2,6,5,8,4

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

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

%C We have verified this for n up to 10^8.

%C We also have some similar conjectures, for example, any integer n > 3 not equal to 17 or 66 can be written as p + q with p and (p+1)*q/2 - 1 both prime.

%H Zhi-Wei Sun, <a href="/A230254/b230254.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.

%e a(15) = 1 since 15 = 5 + 10 with 5 and (5+1)*10/2+1 = 31 both prime.

%e a(30) = 1 since 30 = 2 + 28 with 2 and (2+1)*28/2+1 = 43 both prime.

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

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

%Y Cf. A000040, A002375, A219864, A230252, A227908, A227909, A230241.

%K nonn

%O 1,6

%A _Zhi-Wei Sun_, Oct 14 2013