A333621 Numbers that are divisible by the total number of 1's in both the Zeckendorf and the dual Zeckendorf representations of all their divisors (A300837 and A333618).

1, 2, 4, 126, 416, 442, 3025, 4588, 9243, 10428, 11900, 15070, 18176, 19436, 20532, 26956, 28582, 32108, 33028, 35278, 35929, 37634, 47678, 50386, 61952, 69254, 74578, 88984, 93534, 95120, 96334, 100326, 102297, 142894, 144039, 145768, 147664, 152817, 163125, 183002

Offset: 1

Author

Amiram Eldar, Mar 29 2020

Keywords

• nonn
• base

Examples

			126 is a term since A300837(126) = 21 and A333618(126) = 7 are both divisors of 126.
		

Crossrefs

Intersection of A333619 and A333620.

Cf. A007895, A112310, A300837, A330711, A333617, A333618.

Programs

Mathematica

zeckDigSum[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5] * # + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]];
zeckDivDigSum[n_] := DivisorSum[n, zeckDigSum[#] &];
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]]]]];
dualZeckDivDigSum[n_] := DivisorSum[n, dualZeckSum[#] &];
Select[Range[10^4], Divisible[#, zeckDivDigSum[#]] && Divisible[#, dualZeckDivDigSum[#]] &]