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A341882 Number of ordered factorizations of n into 6 factors > 1.

%I A341882 #13 Feb 16 2025 08:34:01
%S A341882 1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,
%T A341882 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,
%U A341882 0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6
%N A341882 Number of ordered factorizations of n into 6 factors > 1.
%H A341882 Alois P. Heinz, <a href="/A341882/b341882.txt">Table of n, a(n) for n = 64..20000</a>
%H A341882 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OrderedFactorization.html">Ordered Factorization</a>
%F A341882 Dirichlet g.f.: (zeta(s) - 1)^6.
%F A341882 a(n) = 15 * A000005(n) - 20 * A007425(n) + 15 * A007426(n) - 6 * A061200(n) + A034695(n) - 6 for n > 1.
%p A341882 b:= proc(n) option remember; series(x*(1+add(b(n/d),
%p A341882       d=numtheory[divisors](n) minus {1, n})), x, 7)
%p A341882     end:
%p A341882 a:= n-> coeff(b(n), x, 6):
%p A341882 seq(a(n), n=64..160);  # _Alois P. Heinz_, Feb 22 2021
%t A341882 b[n_] := b[n] = Series[x*(1 + Sum[b[n/d],
%t A341882      {d, Divisors[n]~Complement~{1, n}}]), {x, 0, 7}];
%t A341882 a[n_] := Coefficient[b[n], x, 6];
%t A341882 Table[a[n], {n, 64, 160}] (* _Jean-François Alcover_, Feb 28 2022, after _Alois P. Heinz_ *)
%Y A341882 Column k=6 of A251683.
%Y A341882 Cf. A000005, A007425, A007426, A034695, A061200, A070824, A074206, A200221, A341880, A341881.
%K A341882 nonn
%O A341882 64,33
%A A341882 _Ilya Gutkovskiy_, Feb 22 2021