A331107 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).

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, 42, 28, 40, 30, 72, 32, 33, 48, 54, 48, 50, 38, 60, 56, 54, 42, 96, 44, 60, 60, 72, 48, 108, 50, 78, 72, 70, 54, 84, 72, 72, 80, 90, 60, 120, 62, 96, 80, 99, 84, 144, 68

Offset: 1

Author

Amiram Eldar, Jan 09 2020

Keywords

• nonn
• mult

Comments

First differs from A034448 at n = 16.

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

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

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

Programs

Mathematica

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 *)

Formula

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