A238509 a(n) = |{0 < k < n: p(n) + p(k) - 1 is prime}|, where p(.) is the partition function (A000041).

0, 1, 1, 2, 2, 3, 2, 1, 1, 2, 1, 2, 3, 4, 1, 4, 5, 2, 1, 5, 2, 1, 1, 3, 5, 2, 3, 2, 2, 4, 7, 3, 2, 2, 5, 6, 3, 7, 3, 3, 4, 3, 3, 2, 2, 4, 7, 4, 8, 3, 9, 4, 6, 4, 3, 7, 3, 2, 3, 4, 5, 3, 7, 4, 3, 5, 1, 9, 10, 6, 8, 2, 3, 3, 6, 6, 3, 1, 2, 7, 1, 6, 5, 2, 6, 8, 3, 4, 1, 1, 1, 9, 12, 3, 2, 3, 8, 4, 3, 2

Offset: 1

Author

Zhi-Wei Sun, Feb 27 2014

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 1. Also, if n > 2 is different from 8 and 25, then p(n) + p(k) + 1 is prime for some 0 < k < n.

(ii) If n > 7, then n + p(k) is prime for some 0 < k < n.

(iii) If n > 1, then p(k) + q(n) is prime for some 0 < k < n, where q(.) is the strict partition function given by A000009. If n > 2, then p(k) + q(n) - 1 is prime for some 0 < k < n. If n > 1 is not equal to 8, then p(k) + q(n) + 1 is prime for some 0 < k < n.

Examples

			a(11) = 1 since p(11) + p(10) - 1 = 56 + 42 - 1 = 97 is prime.
a(247) = 1 since p(247) + p(228) - 1 = 182973889854026 + 40718063627362 - 1 = 223691953481387 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, A232504, A238457.

Programs

Mathematica

p[n_,k_]:=PrimeQ[PartitionsP[n]+PartitionsP[k]-1]
a[n_]:=Sum[If[p[n,k],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]