A333705 Numbers k such that the total number of 1's in the dual Zeckendorf representation of the first k integers is a multiple of k.

1, 2, 8, 21, 100, 204, 401, 3062, 5974, 11402, 22597, 22598, 43553, 85519, 166243, 1218380, 8854646, 248592083, 248592084, 485966511

Offset

1,2

Comments

The corresponding quotients are 1, 1, 2, 3, 5, 6, 7, 10, 11, ...

No more terms below 3*10^9.

Links

Table of n, a(n) for n=1..20.

Example

8 is a term since the numbers 1, 2, ... 8 in the dual Zeckendorf representation are 1, 10, 11, 101, 110, 111, 1010, 1011, and the sum of their numbers of digits of 1 is 1 + 1 + 2 + 2 + 2 + 3 + 2 + 3 = 16 which is divisible by 8.

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]]]]];
seq = {}; sum = 0; Do[sum += dualZeckSum[n]; If[Divisible[sum, n], AppendTo[seq, n]], {n, 1, 10^6}]; seq

Crossrefs

Cf. A095376, A104326, A112310, A114136, A333702, A333703, A333704.

Sequence in context: A107361 A185309 A318040 * A085077 A272827 A153827

Adjacent sequences: A333702 A333703 A333704 * A333706 A333707 A333708

Keyword

nonn,more

Author

Amiram Eldar, Apr 02 2020

Status

approved