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%I #18 Sep 21 2023 15:35:52
%S 0,0,0,0,1,1,1,1,2,2,2,1,3,1,4,2,1,4,2,2,4,2,3,2,4,3,4,4,2,2,2,1,5,3,
%T 4,3,3,2,3,4,2,2,4,2,4,4,1,5,3,2,6,4,1,5,6,3,3,5,1,5,5,2,7,5,3,4,4,3,
%U 4,6,3,4,6,4,5,6,3,7,4,2,6,1,3,5,9,3,3,7,4,3,7,1,6,5,5,5,6,3,6,7
%N Number of primes p < n with 3*p + 8 and (p-1)*n + 1 both prime.
%C Conjecture: a(n) > 0 for all n > 4.
%C This implies A. Murthy's conjecture (cf. A034693) that for any integer n > 1, there is a positive integer k < n such that k*n + 1 is prime.
%C Conjecture verified for n up to 10^9. - _Mauro Fiorentini_, Sep 21 2023
%H Zhi-Wei Sun, <a href="/A230243/b230243.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 [math.NT], 2012-2017.
%e a(8) = 1 since 8 = 3 + 5 with 3, 3*3+8 = 17, (3-1)*8+1 = 17 all prime.
%e a(17) = 1 since 17 = 7 + 10, and 7, 3*7+8 = 29, (7-1)*17+1 = 103 are all prime.
%t a[n_]:=Sum[If[PrimeQ[3Prime[i]+8]&&PrimeQ[(Prime[i]-1)n+1],1,0],{i,1,PrimePi[n-1]}]
%t Table[a[n],{n,1,100}]
%Y Cf. A034693, A085053, A086685, A023210, A219864, A230217, A230219, A230241.
%K nonn
%O 1,9
%A _Zhi-Wei Sun_, Oct 13 2013
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