A066882 Number of partitions of n into prime divisors of n.
1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 4, 1, 3, 2, 2, 1, 5, 1, 2, 1, 3, 1, 21, 1, 1, 2, 2, 2, 7, 1, 2, 2, 5, 1, 28, 1, 3, 4, 2, 1, 9, 1, 6, 2, 3, 1, 10, 2, 5, 2, 2, 1, 71, 1, 2, 4, 1, 2, 42, 1, 3, 2, 43, 1, 13, 1, 2, 6, 3, 2, 49, 1, 9, 1, 2, 1, 97, 2, 2, 2, 5, 1, 151, 2, 3, 2, 2, 2, 17, 1, 8
Offset: 0
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..10000
Programs
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Maple
with(numtheory): a:= proc(n) local b, l; l:= sort([factorset(n)[]]): b:= proc(m, i) option remember; `if`(m=0, 1, `if`(i<1, 0, b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i)))) end; forget(b): b(n, nops(l)) end: seq(a(n), n=0..100); # Alois P. Heinz, Feb 05 2014 -
Mathematica
a[0] = 1; a[n_] := SeriesCoefficient[1/Product[1-x^d, {d, FactorInteger[n][[All, 1]]}], {x, 0, n}]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jul 30 2015, after Vladeta Jovovic *) -
Python
from sympy import factorint from functools import cache def A066882(n): @cache def b(m, i): if m == 0: return 1 if i < 0: return 0 return b(m, i-1) + (0 if l[i]>m else b(m-l[i], i)) l = sorted(factorint(n)) return b(n, len(l)-1) print([A066882(n) for n in range(99)]) # Michael S. Branicky, Jan 08 2025 after Alois P. Heinz
Formula
Coefficient of x^n in expansion of 1/Product_{d is prime divisor of n} (1-x^d). - Vladeta Jovovic, Apr 11 2004
Extensions
More terms from Sascha Kurz, Mar 23 2002
Corrected by Vladeta Jovovic, Apr 11 2004