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

Author

Zhi-Wei Sun, Nov 24 2013

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

Cf. A000040, A000041, A232398.

Programs

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}]