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A328940
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Numbers k such that k divides A003754(k+1).
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0
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1, 2, 3, 23, 31, 61, 62, 173075, 259698, 332429, 2147535, 21217059, 72517101
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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Table of n, a(n) for n=1..13.
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A003754, A014417, A104326, A276488.
Sequence in context: A068887 A260128 A220569 * A024764 A213971 A024773
Adjacent sequences: A328937 A328938 A328939 * A328941 A328942 A328943
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KEYWORD
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nonn,more
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AUTHOR
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Amiram Eldar, Oct 31 2019
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STATUS
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approved
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