%I #11 Jan 10 2020 05:25:51

%S 1,3,4,5,6,12,8,9,10,18,12,20,14,24,24,27,18,30,20,30,32,36,24,36,26,

%T 42,28,40,30,72,32,33,48,54,48,50,38,60,56,54,42,96,44,60,60,72,48,

%U 108,50,78,72,70,54,84,72,72,80,90,60,120,62,96,80,99,84,144,68

%N The sum of Zeckendorf-infinitary divisors of n = Product_{i} p(i)^r(i): divisors d = Product_{i} p(i)^s(i), such that the Zeckendorf expansion (A014417) of each s(i) contains only terms that are in the Zeckendorf expansion of r(i).

%C First differs from A034448 at n = 16.

%H Amiram Eldar, <a href="/A331107/b331107.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = Product_{i} (p^s(i) + 1), where s(i) are the terms in the Zeckendorf representation of e (A014417).

%e a(16) = 27 since 16 = 2^4 and the Zeckendorf expansion of 4 is 101, i.e., its Zeckendorf representation is a set with 2 terms: {1, 3}. There are 4 possible exponents of 2: 0, 1, 3 and 4, corresponding to the subsets {}, {1}, {3} and {1, 3}. Thus 16 has 4 Zeckendorf-infinitary divisors: 2^0 = 1, 2^1 = 2, 2^3 = 8, and 2^4 = 16, and their sum is 1 + 2 + 8 + 16 = 27.

%t fb[n_] := Block[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k--]; Fibonacci[1 + Position[Reverse@fr, _?(# == 1 &)]]]; f[p_, e_] := p^fb[e]; a[1] = 1; a[n_] := Times @@ (Flatten@(f @@@ FactorInteger[n]) + 1); Array[a, 100] (* after _Robert G. Wilson v_ at A014417 *)

%Y The number of Zeckendorf-infinitary divisors of n is in A318465.

%Y Cf. A014417, A034448, A049417, A049418, A074847, A097863.

%K nonn,mult

%O 1,2

%A _Amiram Eldar_, Jan 09 2020