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

Author

Zhi-Wei Sun, Feb 27 2014

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

Cf. A000009, A000040, A000041, A185636.

Programs

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}]