A238577 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.

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

Offset: 1

Author

Zhi-Wei Sun, Mar 01 2014

Keywords

• nonn

Comments

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.

(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.

Examples

			a(5) = 1 since p(5)*q(4)*r(4) + 1 = 7*2*3 + 1 = 43 is prime.
a(20) = 1 since p(20)*q(13)*r(13) + 1 = 627*18*83 + 1 = 936739 is prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Crossrefs

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

Programs

Mathematica

p[n_,k_]:=PrimeQ[PartitionsP[n]*PartitionsQ[k]*(PartitionsP[k]-PartitionsQ[k])+1]
a[n_]:=Sum[If[p[n,k],1,0],{k,1,n}]
Table[a[n],{n,1,80}]