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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
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.
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
KEYWORD
nonn,more
AUTHOR
Amiram Eldar, Oct 31 2019
STATUS
approved