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Numbers not divisible by the number of their distinct prime factors.
6

%I #24 Jun 05 2023 19:35:35

%S 1,15,21,33,35,39,45,51,55,57,63,65,69,70,75,77,85,87,91,93,95,99,110,

%T 111,115,117,119,123,129,130,133,135,140,141,143,145,147,153,154,155,

%U 159,161,170,171,175,177,182,183,185,187,189,190,201,203,205,207,209

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

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

%C 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.

%C 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.

%C The asymptotic density of this sequence is 1 (Cooper and Kennedy, 1989). - _Amiram Eldar_, Jul 10 2020

%H Christian N. K. Anderson, <a href="/A185307/b185307.txt">Table of n, a(n) for n = 1..10000</a>

%H Curtis N. Cooper and Robert E. Kennedy, <a href="http://www.jstor.org/stable/2323194">Chebyshev's inequality and natural density</a>, Amer. Math. Monthly, Vol. 96, No. 2 (1989), pp. 118-124.

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

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

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

%o (PARI) isok(n) = iferr(n % omega(n), E, 1); \\ _Michel Marcus_, Jul 10 2020

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

%K nonn

%O 1,2

%A _Christian N. K. Anderson_ and _Kevin L. Schwartz_, Apr 23 2013