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

1,4

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.

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.

Example

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.

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}]

Crossrefs

Cf. A000009, A000040, A000041, A232504, A238457.

Sequence in context: A248081 A259632 A304041 * A353932 A368798 A253141

Adjacent sequences: A238506 A238507 A238508 * A238510 A238511 A238512

Keyword

nonn

Author

Zhi-Wei Sun, Feb 27 2014

Status

approved