A048111 Number of unitary divisors of n (A034444) < number of non-unitary divisors of n (A048105).

16, 32, 36, 48, 64, 72, 80, 81, 96, 100, 108, 112, 128, 144, 160, 162, 176, 180, 192, 196, 200, 208, 216, 224, 225, 240, 243, 252, 256, 272, 288, 300, 304, 320, 324, 336, 352, 360, 368, 384, 392, 396, 400, 405, 416, 432, 441, 448, 450, 464, 468, 480, 484

Offset: 1

Author

Labos Elemer

Keywords

• nonn

Comments

Numbers n that are expressible as a product of 2 "nonsquarefree" numbers (i.e., there are 2 integers x,y in A001694 such that n = xy). - Benoit Cloitre, Jan 01 2003

Also numbers having more than one square divisor > 1: A046951(a(n)) > 2. - Reinhard Zumkeller, Apr 08 2003

The asymptotic density of this sequence is 1 - (6/Pi^2)*(1 + Sum_{n>=1} 1/prime(n)^2) = 1 - A059956 * (1 + A085548) = 0.1171394347594477824... . - Amiram Eldar, Sep 25 2022

Examples

			36 is in the sequence since the number of its unitary divisors, {1, 4, 9, 36} is 4 which is smaller than 5, the number of its non-unitary divisors, {2, 3, 6, 12, 18}.
		

Links

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

Crossrefs

Cf. A000005, A001221, A034444, A048105, A048109, A082293, A013929, A082294, A082295.

Cf. A085548, A059956.

Programs

Mathematica

Select[Range[484], DivisorSigma[0, #] > 2^(PrimeNu[#]+1) &] (* Amiram Eldar, Jun 11 2019 *)

PARI

is(n)=my(f=factor(n)[,2],t); for(i=1,#f,if(f[i]>1, if(t||f[i]>3, return(1), t=1))); 0 \\ Charles R Greathouse IV, Sep 17 2015

Formula

A000005(a(n)) > 2^(1 + A001221(a(n))).