cp's OEIS Frontend

A239207 a(n) = |{0 < k <= n: k*p(n)*q(n)*r(n) - 1 is prime}|, where p(.), q(.) and r(.) are given by A000041, A000009 and A047967 respectively.

%I A239207 #12 Mar 13 2014 11:32:42
%S A239207 0,1,3,3,4,3,2,5,4,4,2,3,2,3,5,6,3,4,2,3,5,4,1,6,2,7,3,5,5,3,5,8,7,1,
%T A239207 7,3,8,5,11,7,7,2,6,7,3,7,7,5,5,9,7,7,4,4,6,5,9,7,8,11,4,5,6,8,5,10,5,
%U A239207 6,9,7,10,6,5,5,10,9,8,3,4,1
%N A239207 a(n) = |{0 < k <= n: k*p(n)*q(n)*r(n) - 1 is prime}|, where p(.), q(.) and r(.) are given by A000041, A000009 and A047967 respectively.
%C A239207 Conjecture: (i) a(n) > 0 for all n > 1. If n > 1 is not equal to 25, then k*p(n)*q(n)*r(n) + 1 is prime for some k = 1, ..., n.
%C A239207 (ii) For any integer n > 1, there is a number k among 1, ..., n with k*p(n)*q(n) - 1 (or k*p(n)*q(n) + 1) prime.
%C A239207 (iii) For each n > 1, there is a positive integer k < n with k*p(n) + 1 (or k*q(n) + 1) prime. If n > 1, then k*p(n) - 1 is prime for some k = 1, ..., n. If n > 2, then k*q(n) - 1 is prime for some 0 < k < n.
%C A239207 We have verified that a(n) > 0 for all n = 2, ..., 83000.
%C A239207 See also A239209 and A239214 for related conjectures.
%H A239207 Zhi-Wei Sun, <a href="/A239207/b239207.txt">Table of n, a(n) for n = 1..10000</a>
%H A239207 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014.
%e A239207 a(2) = 1 since 2*p(2)*q(2)*r(2) - 1 = 2*2*1*1 - 1 = 3 is prime.
%e A239207 a(23) = 1 since 12*p(23)*q(23)*r(23) - 1 = 12*1255*104*1151 - 1 = 1802742239 is prime.
%t A239207 p[n_]:=PartitionsP[n]
%t A239207 q[n_]:=PartitionsQ[n]
%t A239207 f[n_]:=p[n]*q[n]*(p[n]-q[n])
%t A239207 a[n_]:=Sum[If[PrimeQ[k*f[n]-1],1,0],{k,1,n}]
%t A239207 Table[a[n],{n,1,80}]
%Y A239207 Cf. A000009, A000040, A000041, A047967, A238393, A238577, A239209, A239214.
%K A239207 nonn
%O A239207 1,3
%A A239207 _Zhi-Wei Sun_, Mar 12 2014