login
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).
0

%I #6 Mar 29 2020 17:14:31

%S 1,2,4,126,416,442,3025,4588,9243,10428,11900,15070,18176,19436,20532,

%T 26956,28582,32108,33028,35278,35929,37634,47678,50386,61952,69254,

%U 74578,88984,93534,95120,96334,100326,102297,142894,144039,145768,147664,152817,163125,183002

%N 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).

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

%t zeckDigSum[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5] * # + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]];

%t zeckDivDigSum[n_] := DivisorSum[n, zeckDigSum[#] &];

%t 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 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 dualZeckDivDigSum[n_] := DivisorSum[n, dualZeckSum[#] &];

%t Select[Range[10^4], Divisible[#, zeckDivDigSum[#]] && Divisible[#, dualZeckDivDigSum[#]] &]

%Y Intersection of A333619 and A333620.

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

%K nonn,base

%O 1,2

%A _Amiram Eldar_, Mar 29 2020