A238458 Number of primes p < n with 2*P(n-p) + 1 prime, where P(.) is the partition function (A000041).

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

Offset

1,4

Comments

Conjecture: a(n) > 0 for all n > 2. Also, for each n > 3 there is a prime p < n with 2*P(n-p) - 1 prime.

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

See also A238459 for a similar conjecture involving the strict partition function.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Example

a(3) = 1 since 2 and 2*P(3-2) + 1 = 2*1 + 1 = 3 are both prime.
a(41) = 1 since 37 and 2*P(41-37) + 1 = 2*5 + 1 = 11 are both prime.

Mathematica

p[n_, k_]:=PrimeQ[2*PartitionsP[n-Prime[k]]+1]
a[n_]:=Sum[If[p[n, k], 1, 0], {k, 1, PrimePi[n-1]}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A000040, A000041, A237705, A237768, A237769, A238457, A238459.

Sequence in context: A002199 A218829 A237715 * A182744 A308355 A336346

Adjacent sequences: A238455 A238456 A238457 * A238459 A238460 A238461

Keyword

nonn

Author

Zhi-Wei Sun, Feb 27 2014

Status

approved