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%I #6 Mar 18 2018 15:02:59
%S 1,1,1,1,1,3,1,6,1,1,1,20,1,46,3,1,1,232,1,501,1,3,9,2352,1,6,14,6,1,
%T 24442,1,53243,3,16,55,32,1,550863,107,55,1,2616338,1,5701553,6,1,406,
%U 27077005,1,25,9,606,9,280237217,3,1244,1,1839,3185,2900328380,1,6320545915,6248,3,1,7828
%N Number of compositions (ordered partitions) of n into parts having the same number of divisors as n.
%H <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F a(n) = [x^n] 1/(1 - Sum_{d(k) = d(n)} x^k).
%e a(14) = 3 because we have [14], [8, 6] and [6, 8], where 14, 8 and 6 are numbers with 4 divisors.
%p with(numtheory):
%p a:= proc(m) option remember; local k, b; k, b:= tau(m),
%p proc(n) option remember; `if`(n=0, 1,
%p add(`if`(tau(j)=k, b(n-j), 0), j=1..n))
%p end: b(m)
%p end:
%p seq(a(n), n=0..80); # _Alois P. Heinz_, Mar 18 2018
%t Table[SeriesCoefficient[1/(1 - Sum[Boole[DivisorSigma[0, k] == DivisorSigma[0, n]] x^k, {k, 1, n}]), {x, 0, n}], {n, 0, 65}]
%Y Cf. A000005, A300977, A300978, A301332, A301333.
%K nonn
%O 0,6
%A _Ilya Gutkovskiy_, Mar 18 2018
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