A344890 Number of partitions of prime(n) containing a prime number of distinct primes and an arbitrary number of nonprimes.

0, 0, 1, 3, 20, 42, 151, 265, 753, 3006, 4594, 15117, 31576, 45002, 89125, 235501, 589613, 792426, 1871442, 3251293, 4261819, 9403682, 15690192, 33111688, 86520382, 137957345, 173655404, 273492399, 342231447, 532915031, 2380864800, 3601147053, 6628703864

Offset

1,4

Links

Table of n, a(n) for n=1..33.

Formula

a(n) = A344715(A000040(n)).

Example

a(4) = 3 because there are 3 partitions of prime(4)=7 that contain a prime number of primes (not counting repetitions). These partitions are [5,2] (containing 2 primes), [3,2,2] (containing 2 unique primes) and [3,2,1,1] (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(ithprime(n)$2)):
seq(a(n), n=1..33);  # Alois P. Heinz, Nov 14 2021

Mathematica

nterms=22; Table[Total[Map[If[PrimeQ[Count[#, _?PrimeQ]], 1, 0] &, Map[DeleteDuplicates[#]&, IntegerPartitions[Prime[n]], {1}]]], {n, 1, nterms}]

Crossrefs

Cf. A000040, A058698, A085755, A235945, A343753, A344677, A344715.

Sequence in context: A337478 A248818 A268602 * A072472 A267055 A296252

Adjacent sequences: A344887 A344888 A344889 * A344891 A344892 A344893

Keyword

nonn

Author

Paolo Xausa, Jun 01 2021

Extensions

a(23)-a(33) from Alois P. Heinz, Jun 02 2021

Status

approved