

A328940


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


0



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

1,2


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.


LINKS



EXAMPLE

23 is in the sequence since the dual Zeckendorf representation of 23 is 101110 that equals 46 when read as a binary number, and 2346.


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]


CROSSREFS



KEYWORD

nonn,more


AUTHOR



STATUS

approved



