A202650 - OEIS

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

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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.