Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #6 Nov 01 2019 18:35:19
%S 1,2,3,23,31,61,62,173075,259698,332429,2147535,21217059,72517101
%N Numbers k such that k divides A003754(k+1).
%C Numbers that divide the value of their dual Zeckendorf representation (A104326) when read as a binary number.
%C Analogous to A276488, with dual Zeckendorf representation instead of Zeckendorf representation (A014417).
%C 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, ...
%C a(14) > 3*10^9, if it exists.
%e 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.
%t 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];
%t 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]]]];
%t aQ[n_] := Divisible[FromDigits[dz[n], 2], n]; Select[Range[100], aQ]
%t (* after _Robert G. Wilson v_ at A014417 and _Ron Knott_'s Maple code at A104326 *)
%Y Cf. A003754, A014417, A104326, A276488.
%K nonn,more
%O 1,2
%A _Amiram Eldar_, Oct 31 2019