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A239232 a(n) = |{0 < k <= n: p(n+k) + 1 is prime}|, where p(.) is the partition function (A000041). 2
1, 0, 0, 1, 3, 3, 3, 2, 1, 1, 1, 1, 2, 4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 4, 4, 3, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 3, 4, 5, 5, 4, 4, 4, 4, 4, 4, 5, 5, 6, 6, 7, 6, 6, 6, 6, 7, 8, 8, 9, 9, 9, 10 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Conjecture: (i) a(n) > 0 for all n > 3.

(ii) If n > 15, then p(n+k) - 1 is prime for some k = 1, ..., n.

(iii) If n > 38, then p(n+k) is prime for some k = 1, ..., n.

The conjecture implies that there are infinitely many positive integers m with p(m) + 1 (or p(m) - 1, or p(m)) prime.

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(4) = 1 since p(4+4) + 1 = 22 + 1 = 23 is prime.

a(8) = 2 since p(8+1) + 1 = 31 and p(8+2) + 1 = 43 are both prime.

a(11) = 1 since p(11+8) + 1 = 491 is prime.

MATHEMATICA

p[n_]:=PartitionsP[n]

a[n_]:=Sum[If[PrimeQ[p[n+k]+1], 1, 0], {k, 1, n}]

Table[a[n], {n, 1, 80}]

CROSSREFS

Cf. A000040, A000041, A049575, A234470, A234569, A238393, A238457, A238509, A238516, A239207, A239209, A239214.

Sequence in context: A201432 A128210 A215409 * A153012 A275300 A283833

Adjacent sequences:  A239229 A239230 A239231 * A239233 A239234 A239235

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Mar 13 2014

STATUS

approved

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Last modified November 14 12:16 EST 2019. Contains 329113 sequences. (Running on oeis4.)