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A331110 The sum of dual-Zeckendorf-infinitary divisors of n = Product_{i} p(i)^r(i): divisors d = Product_{i} p(i)^s(i), such that the dual Zeckendorf expansion (A104326) of each s(i) contains only terms that are in the dual Zeckendorf expansion of r(i).

%I A331110 #12 Mar 12 2023 00:14:56
%S A331110 1,3,4,5,6,12,8,15,10,18,12,20,14,24,24,27,18,30,20,30,32,36,24,60,26,
%T A331110 42,40,40,30,72,32,45,48,54,48,50,38,60,56,90,42,96,44,60,60,72,48,
%U A331110 108,50,78,72,70,54,120,72,120,80,90,60,120,62,96,80,135,84,144
%N A331110 The sum of dual-Zeckendorf-infinitary divisors of n = Product_{i} p(i)^r(i): divisors d = Product_{i} p(i)^s(i), such that the dual Zeckendorf expansion (A104326) of each s(i) contains only terms that are in the dual Zeckendorf expansion of r(i).
%C A331110 First differs from A188999 at n = 32.
%H A331110 Amiram Eldar, <a href="/A331110/b331110.txt">Table of n, a(n) for n = 1..10000</a>
%F A331110 Multiplicative with a(p^e) = Product_{i} (p^s(i) + 1), where s(i) are the terms in the dual Zeckendorf representation of e (A104326).
%e A331110 a(32) = 45 since 32 = 2^5 and the dual Zeckendorf expansion of 5 is 110, i.e., its dual Zeckendorf representation is a set with 2 terms: {2, 3}. There are 4 possible exponents of 2: 0, 2, 3 and 5, corresponding to the subsets {}, {2}, {3} and {2, 3}. Thus 32 has 4 dual-Zeckendorf-infinitary divisors: 2^0 = 1, 2^2 = 4, 2^3 = 8, and 2^5 = 32, and their sum is 1 + 4 + 8 + 32 = 45.
%t A331110 fibTerms[n_] := Module[{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--]; fr];
%t A331110 dualZeck[n_] := Module[{v = fibTerms[n]}, nv = Length[v]; i = 1; While[i <= nv - 2, If[v[[i]] == 1 && v[[i + 1]] == 0 && v[[i + 2]] == 0, v[[i]] = 0; v[[i + 1]] = 1; v[[i + 2]] = 1; If[i > 2, i -= 3]]; i++]; i = Position[v, _?(# > 0 &)]; If[i == {}, {}, v[[i[[1, 1]] ;; -1]]]];
%t A331110 f[p_, e_] := p^Fibonacci[1 + Position[Reverse@dualZeck[e], _?(# == 1 &)]];
%t A331110 a[1] = 1; a[n_] := Times @@ (Flatten@(f @@@ FactorInteger[n]) + 1); Array[a, 100]
%Y A331110 The number of dual-Zeckendorf-infinitary divisors of n is in A331109.
%Y A331110 Cf. A049417, A049418, A074847, A097863, A104326, A188999, A331107.
%K A331110 nonn,mult
%O A331110 1,2
%A A331110 _Amiram Eldar_, Jan 09 2020