A333703 -id:A333703 - cp's OEIS frontend

A333702 Numbers k such that k divides the sum of digits in factorial base of all numbers from 1 to k.

1, 2, 10, 22, 25, 29, 33, 70, 118, 358, 598, 1438, 1803, 1819, 2878, 2881, 2997, 4318, 4322, 4388, 10078, 20158, 21967, 21971, 21975, 30238, 30241, 30837, 40318, 120958, 141121, 142557, 201598, 214563, 214675, 282238, 362878, 649446, 649504, 1088638, 1303204, 1303314

Offset: 1

Amiram Eldar, Apr 02 2020

The corresponding quotients are 1, 1, 2, 3, 3, 3, 3, 4, 5, ...

			10 is a term since the sum of digits in factorial base (A034968) of k from 1 to 10 is 1 + 1 + 2 + 2 + 3 + 1 + 2 + 2 + 3 + 3 = 20, which is divisible by 10.

Cf. A007623, A034968, A368342.

Cf. A095376, A114136, A333703, A333704, A333705.

f[n_] := Module[{s=0, i=2, k=n}, While[k > 0, k = Floor[n/i!]; s = s + (i-1)*k; i++]; n-s]; seq = {}; s = 0; Do[s += f[n]; If[Divisible[s, n], AppendTo[seq, n]], {n, 1, 10^5}]; seq (* after Jean-François Alcover at A034968 *)

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.

Original entry on oeis.org

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

Amiram Eldar, Apr 02 2020

nonn

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

			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.

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

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

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

More terms from Amiram Eldar, Oct 12 2023

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.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Apr 02 2020

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

No more terms below 3*10^9.

			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.

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

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

cp's OEIS Frontend

A333702 Numbers k such that k divides the sum of digits in factorial base of all numbers from 1 to k.

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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.

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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.

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Author

Keywords

Comments

Examples

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Programs

Mathematica