A300979 - OEIS

%I #9 Jul 09 2018 17:20:38

%S 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,

%T 380,1,520,607,6,12,5,18,1268,17,11,22,2214,1,2898,30,21,38,4872,51,

%U 6267,54,35,63,10219,84,43,91,61,107,20587,2,25793,150,105,35924,112,1,49646,247,180,1

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

%H Alois P. Heinz, <a href="/A300979/b300979.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/(1 - x^k).

%e 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.

%p with(numtheory):

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

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

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

%p       end: b(m$2)

%p     end:

%p seq(a(n), n=0..80);  # _Alois P. Heinz_, Mar 17 2018

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

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

%K nonn

%O 0,5

%A _Ilya Gutkovskiy_, Mar 17 2018