A239675 Least k > 0 such that p(n)+k is prime, where p(n) is the number of partitions of n.

1, 1, 1, 2, 2, 4, 2, 2, 1, 1, 1, 3, 2, 2, 2, 3, 2, 10, 4, 1, 4, 5, 7, 4, 4, 15, 1, 1, 1, 2, 19, 15, 4, 8, 13, 4, 4, 10, 2, 4, 1, 4, 15, 16, 6, 3, 5, 5, 10, 6, 7, 4, 20, 10, 4, 1, 6, 13, 3, 1, 14, 4, 25, 8, 21, 39, 29, 8, 2, 14, 1, 34, 16, 12, 17

Offset: 0

Author

Sean A. Irvine, Mar 23 2014

Keywords

• nonn

Comments

Conjecture of Zhi-Wei Sun: a(n) <= n for n > 0.

Examples

			a(3)=2 because p(3)=3 and p(3)+1=4 is composite, but p(3)+2=5 is prime.
		

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(i).

Crossrefs

Cf. A000009, A000040, A000041, A238457, A239736, A240545.

Programs

Mathematica

a[n_] := a[n] = For[pn = PartitionsP[n]; k = 1, True, k++, If[PrimeQ[pn+k], Return[k]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jan 26 2019 *)

PARI

s=[]; for(n=0, 100, k=1; while(!isprime(numbpart(n)+k), k++); s=concat(s, k)); s \\ Colin Barker, Mar 26 2014