A300980 Number of partitions of n into distinct parts having the same number of distinct prime divisors as n.

1, 1, 1, 1, 1, 2, 1, 3, 2, 4, 1, 5, 1, 6, 1, 1, 10, 9, 2, 12, 2, 2, 2, 19, 3, 24, 3, 30, 4, 36, 1, 43, 48, 4, 6, 4, 8, 73, 8, 7, 9, 103, 1, 121, 12, 11, 15, 162, 17, 187, 20, 17, 21, 247, 28, 22, 30, 27, 32, 371, 1, 423, 43, 41, 512, 47, 1, 614, 66, 65, 1, 781, 90, 879, 98, 99, 109, 109

Offset

0,6

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Index entries for sequences related to partitions

Formula

a(n) = [x^n] Product_{omega(k) = omega(n)} (1 + x^k).

Example

a(18) = 2 because we have [18] and [12, 6], where 18, 12 and 6 are numbers that are divisible by exactly 2 different primes.

Maple

with(numtheory):
a:= proc(m) option remember; local k, b; k, b:= nops(factorset(m)),
      proc(n, i) option remember; `if`(i*(i+1)/2<n, 0, `if`(n=0, 1,
        b(n, i-1)+`if`(nops(factorset(i))=k, b(n-i, min(i-1, n-i)), 0)))
      end: b(m$2)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 17 2018

Mathematica

Table[SeriesCoefficient[Product[(1 + Boole[PrimeNu[k] == PrimeNu[n]] x^k), {k, 1, n}], {x, 0, n}], {n, 0, 77}]

Crossrefs

Cf. A001221, A300977, A300978, A300979, A300982, A300983.

Sequence in context: A243499 A124223 A295908 * A260429 A094193 A278539

Adjacent sequences: A300977 A300978 A300979 * A300981 A300982 A300983

Keyword

nonn

Author

Ilya Gutkovskiy, Mar 17 2018

Status

approved