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

Author

Amiram Eldar, Dec 07 2022

Keywords

• nonn
• base

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.

Examples

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

Links

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

Crossrefs

Cf. A007895, A059956, A063743, A328208.

Subsequences: A000040, A000045.

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

Programs

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 *)

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