A238577 - OEIS

%I #11 Jun 24 2017 01:02:47

%S 0,1,1,2,1,2,4,3,4,3,7,4,5,6,4,4,6,4,7,1,4,6,2,8,6,6,5,4,5,4,8,5,9,3,

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

%U 3,9,4,8,12,5,2,8,8,3,6,10,9,9

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

%C Conjecture: (i) a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 3, 5, 20. If n > 2, then p(n)*q(k)*r(k) - 1 is prime for some k = 1, ..., n.

%C (ii) If n > 2 is not equal to 22, then p(n)*q(n)*q(k) - 1 is prime for some k = 1, ..., n. If n > 13, then p(n)*q(k)*q(n-k) - 1 is prime for some 1 < k < n/2.

%H Zhi-Wei Sun, <a href="/A238577/b238577.txt">Table of n, a(n) for n = 1..10000</a>

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014.

%e a(5) = 1 since p(5)*q(4)*r(4) + 1 = 7*2*3 + 1 = 43 is prime.

%e a(20) = 1 since p(20)*q(13)*r(13) + 1 = 627*18*83 + 1 = 936739 is prime.

%t p[n_,k_]:=PrimeQ[PartitionsP[n]*PartitionsQ[k]*(PartitionsP[k]-PartitionsQ[k])+1]

%t a[n_]:=Sum[If[p[n,k],1,0],{k,1,n}]

%t Table[a[n],{n,1,80}]

%Y Cf. A000009, A000040, A000041, A047967, A236442, A238457, A238509, A238393, A238516.

%K nonn

%O 1,4

%A _Zhi-Wei Sun_, Mar 01 2014