A328940 Numbers k such that k divides A003754(k+1).

1, 2, 3, 23, 31, 61, 62, 173075, 259698, 332429, 2147535, 21217059, 72517101

Offset: 1

Author

Amiram Eldar, Oct 31 2019

Keywords

• nonn
• more

Comments

Numbers that divide the value of their dual Zeckendorf representation (A104326) when read as a binary number.

Analogous to A276488, with dual Zeckendorf representation instead of Zeckendorf representation (A014417).

The corresponding values of A003754(k+1) are 1, 2, 3, 46, 62, 183, 186, 15576750, 28826478, 45542773, 534736215, 15934011309, 100218633582, ... and the corresponding quotients are 1, 1, 1, 2, 2, 3, 3, 90, 111, 137, 249, 751, 1382, ...

a(14) > 3*10^9, if it exists.

Examples

			23 is in the sequence since the dual Zeckendorf representation of 23 is 101110 that equals 46 when read as a binary number, and 23|46.
		

Crossrefs

Cf. A003754, A014417, A104326, A276488.

Programs

Mathematica

fb[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];
dz[n_] := Module[{v = fb[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}, v[[i[[1,1]];;-1]]]];
aQ[n_] := Divisible[FromDigits[dz[n], 2], n]; Select[Range[100], aQ]
(* after Robert G. Wilson v at A014417 and Ron Knott's Maple code at A104326 *)