A239736 -id:A239736 - cp's OEIS frontend

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

Sean A. Irvine, Mar 23 2014

nonn

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

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

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).

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

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 *)

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

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.

Original entry on oeis.org

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

Sean A. Irvine, Mar 29 2014

nonn

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

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

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).

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

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 *)

cp's OEIS Frontend

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

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