A239936 Least k > 0 such that p(k)+q(n) is prime, where p(n) is the number of partitions of n and q(n) is the number of strict partitions of n.

1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 2, 1, 1, 2, 4, 3, 1, 4, 3, 3, 10, 3, 4, 5, 2, 1, 1, 1, 6, 5, 5, 1, 6, 2, 4, 1, 12, 1, 15, 13, 1, 3, 5, 4, 4, 1, 5, 5, 1, 2, 1, 12, 49, 1, 1, 2, 6, 6, 3, 14, 3, 3, 3, 6, 6, 16, 13, 16, 11, 1, 1, 4, 5, 3, 12, 25

Offset: 0

Author

Sean A. Irvine, Mar 29 2014

Keywords

• nonn

Comments

Conjecture of Zhi-Wei Sun: a(n) < n for n > 1.

Examples

			a(5)=2 since q(5)+p(2)=3+2=5 is prime but q(5)+p(1)=4 is composite.
		

Links

• Sean A. Irvine, Table of n, a(n) for n = 0..9999
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014-2016. See Conjecture 4.1(ii).

Crossrefs

Cf. A000009, A000040, A000041, A239675, A239736.

Programs

Mathematica

a[n_] := For[k = 1, True, k++, If[PrimeQ[PartitionsP[k] + PartitionsQ[n]], Return[k]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 19 2019 *)