A185307 Numbers not divisible by the number of their distinct prime factors.

1, 15, 21, 33, 35, 39, 45, 51, 55, 57, 63, 65, 69, 70, 75, 77, 85, 87, 91, 93, 95, 99, 110, 111, 115, 117, 119, 123, 129, 130, 133, 135, 140, 141, 143, 145, 147, 153, 154, 155, 159, 161, 170, 171, 175, 177, 182, 183, 185, 187, 189, 190, 201, 203, 205, 207, 209

Offset: 1

Author

Christian N. K. Anderson and Kevin L. Schwartz, Apr 23 2013

Keywords

• nonn

Comments

The complement of A075592 (omega(n) divides n).

Though initially sparse, the sequence increases in density. There are more numbers divisible by omega(n) than not from [3,9265], but there are always more indivisible numbers thereafter.

There are 308 more numbers divisible than indivisible in the range from 1 to 2754, 2778, and 2880. This three values are the global maxima.

The asymptotic density of this sequence is 1 (Cooper and Kennedy, 1989). - Amiram Eldar, Jul 10 2020

Examples

			The distinct prime factors of 45 are 3 and 5, but 45 is not divisible by 2.
		

Links

• Christian N. K. Anderson, Table of n, a(n) for n = 1..10000
• Curtis N. Cooper and Robert E. Kennedy, Chebyshev's inequality and natural density, Amer. Math. Monthly, Vol. 96, No. 2 (1989), pp. 118-124.

Crossrefs

Cf. A075592 (complement), A001221, A001222, A074946, A134334.

Programs

Mathematica

Join[{1},Select[Range[2,300],Mod[#,PrimeNu[#]]!=0&]] (* Harvey P. Dale, Jun 05 2023 *)

PARI

isok(n) = iferr(n % omega(n), E, 1); \\ Michel Marcus, Jul 10 2020

R

library(numbers); isint<-function(x) x==as.integer(x); which(!vapply(1:500,function(n) isint(n/omega(n)),T))