A358978 Numbers that are coprime to the number of terms in their Zeckendorf representation (A007895).

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, 39, 40, 41, 43, 44, 47, 49, 50, 51, 52, 53, 55, 57, 59, 61, 62, 63, 64, 65, 67, 70, 71, 73, 75, 77, 79, 83, 85, 87, 88, 89, 91, 95, 97, 98, 100, 101, 103, 104, 107, 109

Offset

1,2

Comments

First differs from A063743 at n = 22.

Numbers k such that gcd(k, A007895(k)) = 1.

The Fibonacci numbers (A000045) are terms. These are also the only Zeckendorf-Niven numbers (A328208) in this sequence.

Includes all the prime numbers.

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.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

3 is a term since A007895(3) = 1, and gcd(3, 1) = 1.

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A007895, A059956, A063743, A328208.

Subsequences: A000040, A000045.

Similar sequences: A094387, A339076, A358975, A358976, A358977.

Sequence in context: A256133 A078643 A137698 * A063743 A353968 A144100

Adjacent sequences: A358975 A358976 A358977 * A358979 A358980 A358981

Keyword

nonn,base

Author

Amiram Eldar, Dec 07 2022

Status

approved