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 A238457 a(n) = |{0 < k <= n: p(n) + k is prime}|, where p(.) is the partition function (A000041). 12
 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 4, 3, 3, 3, 3, 3, 4, 4, 3, 4, 2, 3, 2, 4, 4, 4, 3, 1, 4, 4, 4, 3, 3, 4, 4, 5, 4, 5, 4, 3, 3, 5, 2, 2, 8, 6, 6, 2, 4, 5, 6, 3, 7, 6, 4, 6, 5, 6, 4, 3, 3, 4, 2, 4, 5, 7, 5, 6, 4, 7, 7, 5, 2, 5, 6, 2, 6, 5, 4, 7, 7, 5, 6, 5, 3, 6, 2, 6, 4, 9, 8, 2, 5, 7, 6, 4, 2, 8 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 2, 3, 4, 5, 30, 109. Also, for each n > 2 there is a positive integer k <= n+3 such that p(n) - k is prime. (ii) For the strict partition function q(.) given by A000009, we have |{0 < k <= n: q(n) + k is prime}| > 0 for all n > 0 and |{0 < k <= n: q(n) - k is prime}| > 0 for all n > 4. 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(5) = 1 since p(5) + 4 = 7 + 4 = 11 is prime. a(30) = 1 since p(30) + 19 = 5604 + 19 = 5623 is prime. a(109) = 1 since p(109) + 63 = 541946240 + 63 = 541946303 is prime. MATHEMATICA p[n_, k_]:=PrimeQ[PartitionsP[n]+k] a[n_]:=Sum[If[p[n, k], 1, 0], {k, 1, n}] Table[a[n], {n, 1, 100}] CROSSREFS Cf. A000009, A000040, A000041, A185636. Sequence in context: A141053 A301507 A005861 * A291310 A304884 A286707 Adjacent sequences:  A238454 A238455 A238456 * A238458 A238459 A238460 KEYWORD nonn AUTHOR Zhi-Wei Sun, Feb 27 2014 STATUS approved

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Last modified March 29 15:16 EDT 2020. Contains 333107 sequences. (Running on oeis4.)