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

0,8

Links

Table of n, a(n) for n=0..49.

Example

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).

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}]

Prog

(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

Crossrefs

Cf. A000040, A058698, A085755, A235945, A343753, A344677, A344890.

Sequence in context: A265702 A190310 A046746 * A058599 A238662 A365272

Adjacent sequences: A344712 A344713 A344714 * A344716 A344717 A344718

Keyword

nonn

Author

Paolo Xausa, May 27 2021

Status

approved