A238393 a(n) = |{0 < k <= n: 2*p(k)*p(n) + 1 is prime}|, where p(.) is the partition function (A000041).

1, 1, 3, 2, 3, 2, 5, 2, 6, 3, 3, 5, 1, 9, 3, 4, 5, 5, 6, 2, 7, 3, 5, 8, 3, 4, 8, 10, 7, 10, 6, 7, 9, 8, 8, 6, 6, 4, 12, 10, 10, 8, 6, 6, 5, 7, 8, 6, 10, 5, 9, 9, 11, 7, 7, 6, 9, 11, 8, 7, 11, 6, 9, 8, 4, 8, 5, 18, 14, 10, 9, 7, 8, 6, 13, 9, 4, 7, 7, 15

Offset: 1

Author

Zhi-Wei Sun, Mar 01 2014

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 2, 13.

(ii) If n > 1, then 2*p(k)*p(n) - 1 is prime for some 0 < k < n.

(iii) For any integer n > 0, p(k)*(p(n)+1) + 1 is prime for some k = 1, ..., n.

Examples

			a(2) = 1 since 2*p(1)*p(2) + 1 = 2*1*2 + 1 = 5 is prime.
a(13) = 1 since 2*p(3)*p(13) + 1 = 2*3*101 + 1 = 607 is 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. A000040, A000041, A238457, A238509, A238516.

Programs

Mathematica

p[n_,k_]:=PrimeQ[2*PartitionsP[n]*PartitionsP[k]+1]
a[n_]:=Sum[If[p[n,k],1,0],{k,1,n}]
Table[a[n],{n,1,80}]