%I #20 Nov 05 2025 15:22:25

%S 0,0,0,0,0,1,1,2,1,2,2,1,2,3,2,2,4,1,4,5,1,6,2,3,6,3,1,2,6,2,3,7,3,6,

%T 4,2,4,2,5,6,1,2,6,5,4,6,8,3,5,10,3,6,6,2,9,4,2,4,6,3,4,11,1,6,7,2,9,

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

%N Number of ways to write n = p + q with p, 3*p - 10 and (p-1)*q - 1 all prime, where q is a positive integer.

%C Conjecture: a(n) > 0 for all n > 5.

%C This implies A. Murthy's conjecture mentioned in A109909.

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

%C Conjecture verified for n up to 10^9. - _Mauro Fiorentini_, Jul 29 2023

%H Zhi-Wei Sun, <a href="/A230241/b230241.txt">Table of n, a(n) for n = 1..10000</a>

%H Zhi-Wei Sun, <a href="https://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, preprint, arXiv:1211.1588 [math.NT], 2012-2017.

%e a(9) = 1 since 9 = 7 + 2 with 7, 3*7-10 = 11, (7-1)*2-1 = 11 all prime.

%e a(27) = 1 since 27 = 13 + 14, and the three numbers 13, 3*13-10 = 29, (13-1)*14-1 = 167 are prime.

%t a[n_]:=Sum[If[PrimeQ[3Prime[i]-10]&&PrimeQ[(Prime[i]-1)(n-Prime[i])-1],1,0],{i,1,PrimePi[n-1]}]

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

%Y Cf. A109909, A227908, A227909, A230227, A230230.

%K nonn

%O 1,8

%A _Zhi-Wei Sun_, Oct 13 2013