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

0,4

Comments

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

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).

Example

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A126768 A261872 A021450 * A289827 A092188 A340675

Adjacent sequences: A239672 A239673 A239674 * A239676 A239677 A239678

Keyword

nonn

Author

Sean A. Irvine, Mar 23 2014

Status

approved