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A341880 Number of ordered factorizations of n into 4 factors > 1.

%I A341880 #14 Feb 16 2025 08:34:01
%S A341880 1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,6,0,0,0,4,0,0,0,0,0,0,0,16,0,
%T A341880 0,0,0,0,4,0,4,0,0,0,12,0,0,0,10,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,16,1,
%U A341880 0,0,12,0,0,0,4,0,12,0,0,0,0,0,40,0,0,0,6,0,0,0,4,0,0,0,28,0,0,0,16
%N A341880 Number of ordered factorizations of n into 4 factors > 1.
%H A341880 Alois P. Heinz, <a href="/A341880/b341880.txt">Table of n, a(n) for n = 16..20000</a>
%H A341880 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OrderedFactorization.html">Ordered Factorization</a>
%F A341880 Dirichlet g.f.: (zeta(s) - 1)^4.
%F A341880 a(n) = 6 * A000005(n) - 4 * A007425(n) + A007426(n) - 4 for n > 1.
%p A341880 b:= proc(n) option remember; series(x*(1+add(b(n/d),
%p A341880       d=numtheory[divisors](n) minus {1, n})), x, 5)
%p A341880     end:
%p A341880 a:= n-> coeff(b(n), x, 4):
%p A341880 seq(a(n), n=16..112);  # _Alois P. Heinz_, Feb 22 2021
%t A341880 b[n_] := b[n] = Series[x*(1 + Sum[b[n/d],
%t A341880      {d, Divisors[n] ~Complement~ {1, n}}]), {x, 0, 5}];
%t A341880 a[n_] := Coefficient[b[n], x, 4];
%t A341880 Table[a[n], {n, 16, 112}] (* _Jean-François Alcover_, Feb 28 2022, after _Alois P. Heinz_ *)
%Y A341880 Column k=4 of A251683.
%Y A341880 Cf. A000005, A007425, A007426, A070824, A074206, A200221, A341881, A341882.
%K A341880 nonn
%O A341880 16,9
%A A341880 _Ilya Gutkovskiy_, Feb 22 2021