A333704 Numbers k such that the total number of 1's in the Zeckendorf representation of the first k integers is a multiple of k.

1, 2, 3, 28, 29, 1119, 6133, 6134, 1141774, 6851892, 6854270, 6854271, 6880561, 219181118, 1113539751, 1187863323, 1200376103, 1247070050, 1247070068, 1247070100, 1247070104, 1247070130, 1251287495, 1252760510, 1257001167, 40920315565, 41404469929, 41473080530

Offset: 1

Author

Amiram Eldar, Apr 02 2020

Keywords

• nonn

Comments

The corresponding quotients are 1, 1, 1, 2, 2, 4, 5, 5, 8, ...

Examples

			3 is a term since the numbers 1, 2 and 3 in the Zeckendorf representation are 1, 10 and 100, and the sum of their numbers of digits of 1 is 1 + 1 + 1 = 3 which is divisible by 3.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..85 (terms below 10^13)

Crossrefs

Cf. A007895, A014417, A095376, A114136, A333702, A333703, A333705.

Programs

Mathematica

zeckSum[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; seq = {}; sum = 0; Do[sum += zeckSum[n]; If[Divisible[sum, n], AppendTo[seq, n]], {n, 1, 10^6}]; seq

Extensions

More terms from Amiram Eldar, Oct 12 2023