A300983 Number of partitions of n into distinct parts having the same number of prime divisors (counted with multiplicity) as n.

1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 2, 1, 3, 2, 2, 1, 5, 1, 5, 2, 1, 2, 7, 2, 9, 1, 4, 6, 9, 1, 11, 6, 10, 2, 14, 2, 15, 1, 2, 12, 19, 1, 18, 5, 16, 2, 26, 1, 24, 2, 23, 26, 35, 2, 39, 31, 3, 1, 40, 4, 50, 6, 49, 9, 61, 1, 67, 68, 7, 6, 70, 10, 87, 2, 1, 93, 102, 2, 107

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_{bigomega(k) = bigomega(n)} (1 + x^k).

Example

a(20) = 2 because we have [20] and [12, 8], where 20, 12 and 8 are numbers that are the product of exactly 3 (not necessarily distinct) primes.

Maple

with(numtheory):
a:= proc(m) option remember; local k, b; k, b:= bigomega(m),
      proc(n, i) option remember; `if`(i*(i+1)/2<n, 0, `if`(n=0, 1,
        b(n, i-1)+`if`(bigomega(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[PrimeOmega[k] == PrimeOmega[n]] x^k), {k, 1, n}], {x, 0, n}], {n, 0, 85}]

Crossrefs

Cf. A001222, A300977, A300978, A300979, A300980, A300982.

Sequence in context: A245222 A022300 A347552 * A279205 A105690 A214364

Adjacent sequences: A300980 A300981 A300982 * A300984 A300985 A300986

Keyword

nonn

Author

Ilya Gutkovskiy, Mar 17 2018

Status

approved