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.

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

Author

Charles R Greathouse IV, Aug 03 2016

Keywords

• nonn

Comments

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

References

• 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.

Links

• 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.

Crossrefs

Cf. A063743, A001221.

Programs

Mathematica

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

PARI

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