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

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

Cf. A000009, A000040, A000041, A185636.

Sequence in context: A141053 A301507 A005861 * A291310 A336014 A304884

Adjacent sequences: A238454 A238455 A238456 * A238458 A238459 A238460

Keyword

nonn

Author

Zhi-Wei Sun, Feb 27 2014

Status

approved