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Number of compositions (ordered partitions) of n into parts having the same number of divisors as n.
2

%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