A063743 -id:A063743 - cp's OEIS frontend

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

Amiram Eldar, Dec 07 2022

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.

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

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

Cf. A007895, A059956, A063743, A328208.
Subsequences: A000040, A000045.
Similar sequences: A094387, A339076, A358975, A358976, A358977.

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 *)

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

A275616 Numbers n such that n and omega(n) are relatively prime, where omega(n) (A001221) is the number of distinct prime divisors of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 64, 65, 67, 69, 70, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 107, 109, 110, 111, 113, 115, 117, 119, 121, 123, 125, 127, 128, 129, 130, 131, 133, 135

Offset: 1

Charles R Greathouse IV, Aug 03 2016

nonn

Alladi shows that the density of A063743 is 6/Pi^2, and mentions (p. 229) that a slight modification of the proof shows that the density of this sequence is the same, hence a(n) ~ (Pi^2/6)n.

Vol'kovič (1976) proved that the asymptotic density of this sequence is 6/Pi^2. - Amiram Eldar, Jul 10 2020

József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, chapter V, p. 174.
V. E. Vol'kovič, Numbers that are relatively prime to their number of prime divisors (in Russian), Izv. Akad. Nauk USSR Ser. Fiz.-Math. Nauk, Vol. 86, No. 4 (1976), pp. 3-7.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Krishnaswami Alladi, On the probability that n and Omega(n) are relatively prime, Fibonacci Quarterly 19:3 (1981), pp. 228-232.

Cf. A063743, A001221.

Select[Range[200],CoprimeQ[#,PrimeNu[#]]&] (* Harvey P. Dale, Dec 20 2021 *)

PARI
```
is(n)=gcd(omega(n),n)==1
```

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI