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

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, 2, 15, 5, 2, 5, 4323, 5, 5, 5, 203, 2, 296, 1, 52, 52, 5, 1, 32653, 5, 135, 5, 15, 2, 1315, 15, 566, 52, 5, 2, 270920, 2, 5, 52, 203, 5, 296, 5, 52, 52, 877, 2

Offset: 1

Author

Alois P. Heinz, Jul 19 2021

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..119

Crossrefs

Cf. A000045, A001055, A001605, A046523, A072381, A278245.

Programs

Maple

b:= proc(n, k) option remember; `if`(n>k, 0, 1)+`if`(isprime(n), 0,
      add(`if`(d>k, 0, b(n/d, d)), d=numtheory[divisors](n) minus {1, n}))
    end:
a:= proc(n) option remember; b((l-> mul(ithprime(i)^l[i], i=1..nops(l)))(
      sort(map(i-> i[2], ifactors(combinat[fibonacci](n))[2]), `>`))$2)
    end:
seq(a(n), n=1..80);

Mathematica

T[, 1] = T[1, ] = 1;
T[n_, m_] := T[n, m] = DivisorSum[n, If[1 < # <= m, T[n/#, #], 0]&];
f[n_] := T[n, n];
a[n_] := f[Fibonacci[n]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 119}] (* Jean-François Alcover, Sep 08 2022 *)

Formula

a(n) = A001055(A000045).

a(n) = A001055(A046523(A000045(n))).

a(n) = A001055(A278245(n)).

a(n) = 1 <=> n in { A001605 } union {1,2}.

a(n) = 2 <=> n in { A072381 }.