A300979 Number of partitions of n into parts having the same number of distinct prime divisors as n.

1, 1, 1, 1, 2, 2, 1, 4, 6, 7, 1, 12, 2, 19, 1, 1, 37, 44, 3, 66, 3, 2, 3, 138, 6, 196, 5, 275, 6, 380, 1, 520, 607, 6, 12, 5, 18, 1268, 17, 11, 22, 2214, 1, 2898, 30, 21, 38, 4872, 51, 6267, 54, 35, 63, 10219, 84, 43, 91, 61, 107, 20587, 2, 25793, 150, 105, 35924, 112, 1, 49646, 247, 180, 1

Offset

0,5

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/(1 - x^k).

Example

a(18) = 3 because we have [18], [12, 6] and [6, 6, 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`(n=0, 1, `if`(i<1, 0,
        b(n, i-1)+`if`(nops(factorset(i))=k, b(n-i, min(i, n-i)), 0)))
      end: b(m$2)
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Mar 17 2018

Mathematica

Table[SeriesCoefficient[Product[1/(1 - Boole[PrimeNu[k] == PrimeNu[n]] x^k), {k, 1, n}], {x, 0, n}], {n, 0, 70}]

Crossrefs

Cf. A001221, A300977, A300978, A300980, A300982, A300983.

Sequence in context: A053495 A096747 A299504 * A371889 A167622 A084606

Adjacent sequences: A300976 A300977 A300978 * A300980 A300981 A300982

Keyword

nonn

Author

Ilya Gutkovskiy, Mar 17 2018

Status

approved