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A328940 Numbers k such that k divides A003754(k+1). 0
1, 2, 3, 23, 31, 61, 62, 173075, 259698, 332429, 2147535, 21217059, 72517101 (list; graph; refs; listen; history; text; internal format)
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 23|46.
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 *)
CROSSREFS
Sequence in context: A364142 A260128 A220569 * A024764 A213971 A024773
KEYWORD
nonn,more
AUTHOR
Amiram Eldar, Oct 31 2019
STATUS
approved

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Last modified May 27 15:55 EDT 2024. Contains 372880 sequences. (Running on oeis4.)