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

1,3

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.

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.

Example

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.

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}]

Crossrefs

Cf. A000040, A000041, A238457, A238509, A238516.

Sequence in context: A134267 A249800 A165258 * A092962 A258778 A112924

Adjacent sequences: A238390 A238391 A238392 * A238394 A238395 A238396

Keyword

nonn

Author

Zhi-Wei Sun, Mar 01 2014

Status

approved