A344715 Number of partitions of n containing a prime number of distinct primes and an arbitrary number of nonprimes.
0, 0, 0, 0, 0, 1, 1, 3, 5, 9, 12, 20, 27, 42, 56, 80, 107, 151, 195, 265, 342, 453, 577, 753, 949, 1220, 1525, 1930, 2398, 3006, 3701, 4594, 5625, 6922, 8426, 10291, 12455, 15117, 18203, 21955, 26326, 31576, 37689, 45002, 53498, 63581, 75313, 89125, 105199, 124056
Offset: 0
Keywords
Examples
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).
Programs
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Maple
b:= proc(n, i) option remember; expand( `if`(n=0 or i=1, 1, b(n, i-1)+`if`(isprime(i), x, 1) *add(b(n-i*j, i-1), j=1..n/i))) end: a:= n-> (p-> add(`if`(isprime(i), coeff(p, x, i), 0), i=2..degree(p)))(b(n$2)): seq(a(n), n=0..49); # Alois P. Heinz, Nov 14 2021 -
Mathematica
nterms=50;Table[Total[Map[If[PrimeQ[Count[#, _?PrimeQ]],1,0] &,Map[DeleteDuplicates[#]&,IntegerPartitions[n],{1}]]],{n,0,nterms-1}] -
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