A238577 - OEIS

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

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OFFSET

1,4

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.

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.

EXAMPLE

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.

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}]

CROSSREFS

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