A300980 - OEIS

%I #10 Jul 09 2018 17:32:47

%S 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,

%T 43,48,4,6,4,8,73,8,7,9,103,1,121,12,11,15,162,17,187,20,17,21,247,28,

%U 22,30,27,32,371,1,423,43,41,512,47,1,614,66,65,1,781,90,879,98,99,109,109

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

%H Alois P. Heinz, <a href="/A300980/b300980.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

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

%e 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.

%p with(numtheory):

%p a:= proc(m) option remember; local k, b; k, b:= nops(factorset(m)),

%p       proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,`if`(n=0, 1,

%p         b(n, i-1)+`if`(nops(factorset(i))=k, b(n-i, min(i-1, n-i)), 0)))

%p       end: b(m$2)

%p     end:

%p seq(a(n), n=0..100);  # _Alois P. Heinz_, Mar 17 2018

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

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

%K nonn

%O 0,6

%A _Ilya Gutkovskiy_, Mar 17 2018