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

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, 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, 22, 13, 13, 6, 33, 7, 12, 18, 23, 12, 26, 6, 17, 13, 23, 7, 37, 7, 14

Offset: 1

Author

Amiram Eldar, Mar 29 2020

Keywords

• nonn
• base

Examples

			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.
		

Links

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

Crossrefs

Cf. A034690, A093653, A104326, A112310, A300837.

Programs

Mathematica

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];
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]]]]];
a[n_] := DivisorSum[n, dualZeckSum[#] &]; Array[a, 100]

Formula

a(n) = Sum_{d|n} A112310(d).