A300983 - OEIS

%I #10 Jul 09 2018 19:22:39

%S 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,

%T 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,

%U 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

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

%H Alois P. Heinz, <a href="/A300983/b300983.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_{bigomega(k) = bigomega(n)} (1 + x^k).

%e 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.

%p with(numtheory):

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

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

%p         b(n, i-1)+`if`(bigomega(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[PrimeOmega[k] == PrimeOmega[n]] x^k), {k, 1, n}], {x, 0, n}], {n, 0, 85}]

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

%K nonn

%O 0,6

%A _Ilya Gutkovskiy_, Mar 17 2018