A344677 - OEIS

%I #25 May 31 2021 21:46:08

%S 0,0,0,0,1,2,4,6,9,13,20,26,36,49,68,90,120,154,201,258,330,418,532,

%T 666,834,1041,1290,1592,1958,2404,2935,3588,4345,5278,6366,7692,9215,

%U 11096,13230,15853,18831,22477,26580,31620,37247,44145,51851,61247,71681,84445

%N Number of partitions of n containing a prime number of primes and an arbitrary number of nonprimes.

%H Andrew Howroyd, <a href="/A344677/b344677.txt">Table of n, a(n) for n = 0..1000</a>

%e 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].

%t nterms=50;Table[Total[Map[If[PrimeQ[Count[#, _?PrimeQ]],1,0] &,IntegerPartitions[n]]],{n,0,nterms-1}]

%t (* Second program: *)

%t 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]];

%t seq[50] (* _Jean-François Alcover_, May 27 2021, after _Andrew Howroyd_ *)

%o (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

%Y Cf. A000040, A058698, A085755, A235945, A343753.

%K nonn

%O 0,6

%A _Paolo Xausa_, May 26 2021