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%I #10 Dec 12 2022 01:33:39
%S 1,2,3,5,7,8,9,11,13,15,17,19,20,21,23,25,28,29,31,32,33,34,35,37,38,
%T 39,40,41,43,44,47,49,50,51,52,53,55,57,59,61,62,63,64,65,67,70,71,73,
%U 75,77,79,83,85,87,88,89,91,95,97,98,100,101,103,104,107,109
%N Numbers that are coprime to the number of terms in their Zeckendorf representation (A007895).
%C First differs from A063743 at n = 22.
%C Numbers k such that gcd(k, A007895(k)) = 1.
%C The Fibonacci numbers (A000045) are terms. These are also the only Zeckendorf-Niven numbers (A328208) in this sequence.
%C Includes all the prime numbers.
%C The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 7, 61, 614, 6028, 61226, 606367, 6041106, 61235023, 612542436, 6034626175, 60093287082, 609082612171, ... . Conjecture: The asymptotic density of this sequence exists and equals 6/Pi^2 = 0.607927... (A059956), the same as the density of A094387.
%H Amiram Eldar, <a href="/A358978/b358978.txt">Table of n, a(n) for n = 1..10000</a>
%e 3 is a term since A007895(3) = 1, and gcd(3, 1) = 1.
%t z[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; Select[Range[120], CoprimeQ[#, z[#]] &] (* after _Alonso del Arte_ at A007895 *)
%o (PARI) is(n) = if(n<4, 1, my(k=2, m=n, s, t); while(fibonacci(k++)<=m, ); while(k && m, t=fibonacci(k); if(t<=m, m-=t; s++); k--); gcd(n, s)==1); \\ after _Charles R Greathouse IV_ at A007895
%Y Cf. A007895, A059956, A063743, A328208.
%Y Subsequences: A000040, A000045.
%Y Similar sequences: A094387, A339076, A358975, A358976, A358977.
%K nonn,base
%O 1,2
%A _Amiram Eldar_, Dec 07 2022
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