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A333618 a(n) is the total number of terms (1-digits) in the dual Zeckendorf representation of all divisors of n.

%I A333618 #9 Oct 12 2023 06:20:03
%S A333618 1,2,3,4,3,7,3,7,6,7,5,12,4,8,8,11,5,14,6,12,9,10,5,20,7,9,11,14,6,20,
%T A333618 6,17,11,10,10,23,6,12,11,21,5,22,6,17,17,11,6,30,8,17,13,17,8,23,12,
%U A333618 22,13,13,6,33,7,12,18,23,12,26,6,17,13,23,7,37,7,14
%N A333618 a(n) is the total number of terms (1-digits) in the dual Zeckendorf representation of all divisors of n.
%H A333618 Amiram Eldar, <a href="/A333618/b333618.txt">Table of n, a(n) for n = 1..10000</a>
%F A333618 a(n) = Sum_{d|n} A112310(d).
%e A333618 For n = 6, its divisors are 1, 2, 3 and 6. The dual Zeckendorf representations (A104326) of the divisors are 1, 10, 11 and 111. Their total number of 1's is 1 + 1 + 2 + 3 = 7, thus a(6) = 7.
%t A333618 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 A333618 dualZeckSum[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 == {}, 0, Total[v[[i[[1, 1]] ;; -1]]]]];
%t A333618 a[n_] := DivisorSum[n, dualZeckSum[#] &]; Array[a, 100]
%Y A333618 Cf. A034690, A093653, A104326, A112310, A300837.
%K A333618 nonn,base
%O A333618 1,2
%A A333618 _Amiram Eldar_, Mar 29 2020