A202650 Number of ways to write n = p + p(k) + p(m) with 0 < k <= m, where p is a prime and p(.) is the partition function (A000041).

0, 0, 0, 1, 2, 3, 4, 4, 6, 5, 7, 5, 7, 5, 10, 6, 10, 5, 12, 7, 13, 5, 13, 6, 15, 6, 15, 6, 15, 6, 13, 7, 15, 8, 17, 10, 14, 8, 14, 11, 12, 9, 13, 11, 14, 14, 16, 13, 16, 14, 15, 12, 12, 14, 16, 14, 13, 10, 14, 16, 15, 14, 18, 17, 15, 17, 14, 14, 15, 16, 14, 13, 15, 19, 18, 18, 16, 15, 13, 17, 18, 14, 19, 17, 19, 18, 18, 15, 21, 17, 22, 13, 17, 14, 20, 15, 19, 13, 15, 15

Offset

1,5

Comments

Conjecture: (i) a(n) > 0 for all n > 3.

(ii) For any integer n > 2, |n - p(k)| is prime for some k = 1,...,n. Also, for any positive integer n not equal to 7, n + p(k) is prime for some k = 1,...,n.

We have verified part (i) of the conjecture for all n = 4, 5, ..., 2*10^7.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..3000

Z.-W. Sun, On a^n+ bn modulo m, arXiv preprint arXiv:1312.1166 [math.NT], 2013-2014.

Example

a(6) = 3 since 6 = 3 + p(1) + p(2) = 2 + p(1) + p(3) = 2 + p(2) + p(2) with 2 and 3 prime.

Mathematica

PQ[n_]:=n>1&&PrimeQ[n]
a[n_]:=Sum[If[PQ[n-PartitionsP[m]-PartitionsP[k]], 1, 0], {m, 1, n}, {k, 1, m}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A000040, A000041, A232398.

Sequence in context: A333995 A301764 A181833 * A370829 A228286 A158973

Adjacent sequences: A202647 A202648 A202649 * A202651 A202652 A202653

Keyword

nonn

Author

Zhi-Wei Sun, Nov 24 2013

Status

approved