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 A344677 Number of partitions of n containing a prime number of primes and an arbitrary number of nonprimes. 3
 0, 0, 0, 0, 1, 2, 4, 6, 9, 13, 20, 26, 36, 49, 68, 90, 120, 154, 201, 258, 330, 418, 532, 666, 834, 1041, 1290, 1592, 1958, 2404, 2935, 3588, 4345, 5278, 6366, 7692, 9215, 11096, 13230, 15853, 18831, 22477, 26580, 31620, 37247, 44145, 51851, 61247, 71681, 84445 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 LINKS Andrew Howroyd, Table of n, a(n) for n = 0..1000 EXAMPLE a(6) = 4 because there are 4 partitions of 6 that contain a prime number of primes (including repetitions). These partitions are [3,3], [3,2,1], [2,2,2], [2,2,1,1]. MATHEMATICA nterms=50; Table[Total[Map[If[PrimeQ[Count[#, _?PrimeQ]], 1, 0] &, IntegerPartitions[n]]], {n, 0, nterms-1}] (* Second program: *) seq[n_] := Module[{p}, p = 1/Product[1 - If[PrimeQ[k], y*x^k, 0] + O[x]^n, {k, 2, n}]; CoefficientList[Sum[If[PrimeQ[k], Coefficient[p, y, k], 0], {k, 2, n}]/QPochhammer[x + O[x]^n]/(p /. y -> 1), x]]; seq[50] (* Jean-François Alcover, May 27 2021, after Andrew Howroyd *) PROG (PARI) seq(n)={my(p=1/prod(k=2, n, 1 - if(isprime(k), y*x^k) + O(x*x^n))); Vec(sum(k=2, n, if(isprime(k), polcoef(p, k, y)))/eta(x+O(x*x^n))/subst(p, y, 1), -(n+1))} \\ Andrew Howroyd, May 26 2021 CROSSREFS Cf. A000040, A058698, A085755, A235945, A343753. Sequence in context: A081659 A143586 A363457 * A241546 A280918 A081225 Adjacent sequences: A344674 A344675 A344676 * A344678 A344679 A344680 KEYWORD nonn AUTHOR Paolo Xausa, May 26 2021 STATUS approved

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Last modified February 21 07:26 EST 2024. Contains 370219 sequences. (Running on oeis4.)