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

1,2

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

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.

Example

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

Mathematica

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

Prog

(R) library(numbers); isint<-function(x) x==as.integer(x); which(!vapply(1:500, function(n) isint(n/omega(n)), T))
(PARI) isok(n) = iferr(n % omega(n), E, 1); \\ Michel Marcus, Jul 10 2020

Crossrefs

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

Sequence in context: A184041 A070005 A275384 * A061346 A098905 A225375

Adjacent sequences: A185304 A185305 A185306 * A185308 A185309 A185310

Keyword

nonn

Author

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

Status

approved