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 A238458 Number of primes p < n with 2*P(n-p) + 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 (list; graph; refs; listen; history; text; internal format)
 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(n-p) - 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 Zhi-Wei 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(3-2) + 1 = 2*1 + 1 = 3 are both prime. a(41) = 1 since 37 and 2*P(41-37) + 1 = 2*5 + 1 = 11 are both prime. MATHEMATICA p[n_, k_]:=PrimeQ[2*PartitionsP[n-Prime[k]]+1] a[n_]:=Sum[If[p[n, k], 1, 0], {k, 1, PrimePi[n-1]}] Table[a[n], {n, 1, 100}] CROSSREFS Cf. A000040, A000041, A237705, A237768, A237769, A238457, A238459. Sequence in context: A002199 A218829 A237715 * A182744 A308355 A104324 Adjacent sequences:  A238455 A238456 A238457 * A238459 A238460 A238461 KEYWORD nonn AUTHOR Zhi-Wei Sun, Feb 27 2014 STATUS approved

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Last modified February 18 00:28 EST 2020. Contains 332006 sequences. (Running on oeis4.)