A238459 Number of primes p < n with q(n-p) + 1 prime, where q(.) is the strict partition function (A000009).

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

Offset: 1

Author

Zhi-Wei Sun, Feb 27 2014

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 2. Also, for each n > 6 there is a prime p < n with q(n-p) - 1 prime.

We have verified the conjecture for n up to 10^5.

See also A238458 for a similar conjecture involving the partition function p(n).

Examples

			a(3) = 1 since 2 and q(3-2) + 1 = 1 + 1 = 2 are both prime.
a(28) = 1 since 17 and q(28-17) + 1 = q(11) + 1 = 12 + 1 = 13 are both 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, A237705, A237768, A237769, A238457, A238458.

Programs

Mathematica

q[n_,k_]:=PrimeQ[PartitionsQ[n-Prime[k]]+1]
a[n_]:=Sum[If[q[n,k],1,0],{k,1,PrimePi[n-1]}]
Table[a[n],{n,1,100}]