

A238458


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


3



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
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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(np)  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

ZhiWei 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(32) + 1 = 2*1 + 1 = 3 are both prime.
a(41) = 1 since 37 and 2*P(4137) + 1 = 2*5 + 1 = 11 are both prime.


MATHEMATICA

p[n_, k_]:=PrimeQ[2*PartitionsP[nPrime[k]]+1]
a[n_]:=Sum[If[p[n, k], 1, 0], {k, 1, PrimePi[n1]}]
Table[a[n], {n, 1, 100}]


CROSSREFS

Cf. A000040, A000041, A237705, A237768, A237769, A238457, A238459.
Sequence in context: A002199 A218829 A237715 * A182744 A104324 A193212
Adjacent sequences: A238455 A238456 A238457 * A238459 A238460 A238461


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Feb 27 2014


STATUS

approved



