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A344715 Number of partitions of n containing a prime number of distinct primes and an arbitrary number of nonprimes.

%I A344715 #13 Nov 14 2021 16:03:40
%S A344715 0,0,0,0,0,1,1,3,5,9,12,20,27,42,56,80,107,151,195,265,342,453,577,
%T A344715 753,949,1220,1525,1930,2398,3006,3701,4594,5625,6922,8426,10291,
%U A344715 12455,15117,18203,21955,26326,31576,37689,45002,53498,63581,75313,89125,105199,124056
%N A344715 Number of partitions of n containing a prime number of distinct primes and an arbitrary number of nonprimes.
%e A344715 a(10) = 12 because there are 12 partitions of 10 that contain a prime number of primes (not counting repetitions). These partitions are [7,3] (containing 2 primes), [7,2,1] (containing 2 primes), [5,3,2] (containing 3 primes), [5,3,1,1] (containing 2 primes), [5,2,2,1] (containing 2 distinct primes), [5,2,1,1,1] (containing 2 primes), [4,3,2,1] (containing 2 primes), [3,3,2,2] (containing 2 distinct primes), [3,3,2,1,1] (containing 2 distinct primes), [3,2,2,2,1] (containing 2 distinct primes), [3,2,2,1,1,1] (containing 2 distinct primes) and [3,2,1,1,1,1,1] (containing 2 primes).
%p A344715 b:= proc(n, i) option remember; expand(
%p A344715       `if`(n=0 or i=1, 1, b(n, i-1)+`if`(isprime(i), x, 1)
%p A344715           *add(b(n-i*j, i-1), j=1..n/i)))
%p A344715     end:
%p A344715 a:= n-> (p-> add(`if`(isprime(i), coeff(p, x, i), 0),
%p A344715              i=2..degree(p)))(b(n$2)):
%p A344715 seq(a(n), n=0..49);  # _Alois P. Heinz_, Nov 14 2021
%t A344715 nterms=50;Table[Total[Map[If[PrimeQ[Count[#, _?PrimeQ]],1,0] &,Map[DeleteDuplicates[#]&,IntegerPartitions[n],{1}]]],{n,0,nterms-1}]
%o A344715 (PARI) seq(n)={my(p=prod(k=2, n, 1 - y + y/(1 - if(isprime(k), 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 27 2021
%Y A344715 Cf. A000040, A058698, A085755, A235945, A343753, A344677, A344890.
%K A344715 nonn
%O A344715 0,8
%A A344715 _Paolo Xausa_, May 27 2021