A238457 - OEIS

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A238457

a(n) = |{0 < k <= n: p(n) + k is prime}|, where p(.) is the partition function (A000041).

1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 4, 3, 3, 3, 3, 3, 4, 4, 3, 4, 2, 3, 2, 4, 4, 4, 3, 1, 4, 4, 4, 3, 3, 4, 4, 5, 4, 5, 4, 3, 3, 5, 2, 2, 8, 6, 6, 2, 4, 5, 6, 3, 7, 6, 4, 6, 5, 6, 4, 3, 3, 4, 2, 4, 5, 7, 5, 6, 4, 7, 7, 5, 2, 5, 6, 2, 6, 5, 4, 7, 7, 5, 6, 5, 3, 6, 2, 6, 4, 9, 8, 2, 5, 7, 6, 4, 2, 8

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OFFSET

1,6

COMMENTS

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 2, 3, 4, 5, 30, 109. Also, for each n > 2 there is a positive integer k <= n+3 such that p(n) - k is prime.

(ii) For the strict partition function q(.) given by A000009, we have |{0 < k <= n: q(n) + k is prime}| > 0 for all n > 0 and |{0 < k <= n: q(n) - k is prime}| > 0 for all n > 4.

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, 2014.

EXAMPLE

a(5) = 1 since p(5) + 4 = 7 + 4 = 11 is prime.

a(30) = 1 since p(30) + 19 = 5604 + 19 = 5623 is prime.

a(109) = 1 since p(109) + 63 = 541946240 + 63 = 541946303 is prime.

MATHEMATICA

p[n_, k_]:=PrimeQ[PartitionsP[n]+k]

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

Table[a[n], {n, 1, 100}]

CROSSREFS