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A339211 Zeckendorf self numbers: numbers not of the form k + A007895(k).

%I A339211 #16 Aug 08 2025 03:24:35
%S A339211 1,5,7,10,19,21,27,29,32,36,40,42,45,54,61,63,66,75,77,83,85,88,95,97,
%T A339211 100,109,111,117,119,122,126,130,132,135,144,146,150,152,155,164,166,
%U A339211 172,174,177,181,185,187,190,199,206,208,211,220,222,228,230,233,239
%N A339211 Zeckendorf self numbers: numbers not of the form k + A007895(k).
%C A339211 Analogous to self numbers (A003052) using the Zeckendorf representation (A014417) instead of decimal expansion.
%C A339211 The numbers of terms that do not exceed 10^k, for k = 0, 1, ..., are 1, 4, 25, 236, 2351, 23495, 234949, 2349463, 23494586, 234945839, 2349458364, ... . Apparently, the asymptotic density of this sequence exists and equals 0.23494583... . - _Amiram Eldar_, Aug 08 2025
%D A339211 József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384-386.
%H A339211 Amiram Eldar, <a href="/A339211/b339211.txt">Table of n, a(n) for n = 1..10000</a>
%H A339211 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SelfNumber.html">Self Number</a>.
%H A339211 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ZeckendorfRepresentation.html">Zeckendorf Representation</a>.
%H A339211 Wikipedia, <a href="http://en.wikipedia.org/wiki/Self_number">Self number</a>.
%H A339211 <a href="/index/Coi#Colombian">Index entries for Colombian or self numbers and related sequences</a>
%t A339211 z[n_] := n + Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; m = 250; Complement[Range[m], Array[z, m]] (* after _Alonso del Arte_ at A007895 *)
%Y A339211 Cf. A003052, A007895, A010061, A010064, A010067, A010070, A014417, A328208, A339212, A339213, A339214, A339215.
%K A339211 nonn,base
%O A339211 1,2
%A A339211 _Amiram Eldar_, Nov 27 2020