

A238459


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


3



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
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OFFSET

1,4


COMMENTS

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


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.


EXAMPLE

a(3) = 1 since 2 and q(32) + 1 = 1 + 1 = 2 are both prime.
a(28) = 1 since 17 and q(2817) + 1 = q(11) + 1 = 12 + 1 = 13 are both prime.


MATHEMATICA

q[n_, k_]:=PrimeQ[PartitionsQ[nPrime[k]]+1]
a[n_]:=Sum[If[q[n, k], 1, 0], {k, 1, PrimePi[n1]}]
Table[a[n], {n, 1, 100}]


CROSSREFS

Cf. A000009, A000040, A237705, A237768, A237769, A238457, A238458.
Sequence in context: A080328 A245555 A031266 * A109301 A107573 A081308
Adjacent sequences: A238456 A238457 A238458 * A238460 A238461 A238462


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Feb 27 2014


STATUS

approved



