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A238459 Number of primes p < n with q(n-p) + 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 (list; graph; refs; listen; history; text; internal format)
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(n-p) - 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

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(3) = 1 since 2 and q(3-2) + 1 = 1 + 1 = 2 are both prime.

a(28) = 1 since 17 and q(28-17) + 1 = q(11) + 1 = 12 + 1 = 13 are both prime.

MATHEMATICA

q[n_, k_]:=PrimeQ[PartitionsQ[n-Prime[k]]+1]

a[n_]:=Sum[If[q[n, k], 1, 0], {k, 1, PrimePi[n-1]}]

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

Zhi-Wei Sun, Feb 27 2014

STATUS

approved

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Last modified December 14 12:04 EST 2019. Contains 329979 sequences. (Running on oeis4.)