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A346491 Number of factorizations of the n-th Fibonacci number.

%I A346491 #16 Sep 08 2022 08:29:09
%S A346491 1,1,1,1,1,3,1,2,2,2,1,29,1,2,5,5,1,21,2,15,5,2,1,719,4,2,15,15,1,296,
%T A346491 2,15,5,2,5,4323,5,5,5,203,2,296,1,52,52,5,1,32653,5,135,5,15,2,1315,
%U A346491 15,566,52,5,2,270920,2,5,52,203,5,296,5,52,52,877,2
%N A346491 Number of factorizations of the n-th Fibonacci number.
%H A346491 Alois P. Heinz, <a href="/A346491/b346491.txt">Table of n, a(n) for n = 1..119</a>
%F A346491 a(n) = A001055(A000045).
%F A346491 a(n) = A001055(A046523(A000045(n))).
%F A346491 a(n) = A001055(A278245(n)).
%F A346491 a(n) = 1 <=> n in { A001605 } union {1,2}.
%F A346491 a(n) = 2 <=> n in { A072381 }.
%p A346491 b:= proc(n, k) option remember; `if`(n>k, 0, 1)+`if`(isprime(n), 0,
%p A346491       add(`if`(d>k, 0, b(n/d, d)), d=numtheory[divisors](n) minus {1, n}))
%p A346491     end:
%p A346491 a:= proc(n) option remember; b((l-> mul(ithprime(i)^l[i], i=1..nops(l)))(
%p A346491       sort(map(i-> i[2], ifactors(combinat[fibonacci](n))[2]), `>`))$2)
%p A346491     end:
%p A346491 seq(a(n), n=1..80);
%t A346491 T[_, 1] = T[1, _] = 1;
%t A346491 T[n_, m_] := T[n, m] = DivisorSum[n, If[1 < # <= m, T[n/#, #], 0]&];
%t A346491 f[n_] := T[n, n];
%t A346491 a[n_] := f[Fibonacci[n]];
%t A346491 Table[Print[n, " ", a[n]]; a[n], {n, 1, 119}] (* _Jean-François Alcover_, Sep 08 2022 *)
%Y A346491 Cf. A000045, A001055, A001605, A046523, A072381, A278245.
%K A346491 nonn
%O A346491 1,6
%A A346491 _Alois P. Heinz_, Jul 19 2021