A162643 Numbers whose number of divisors is not a power of 2.

4, 9, 12, 16, 18, 20, 25, 28, 32, 36, 44, 45, 48, 49, 50, 52, 60, 63, 64, 68, 72, 75, 76, 80, 81, 84, 90, 92, 96, 98, 99, 100, 108, 112, 116, 117, 121, 124, 126, 132, 140, 144, 147, 148, 150, 153, 156, 160, 162, 164, 169, 171, 172, 175, 176, 180, 188, 192, 196, 198

Offset: 1

Author

Reinhard Zumkeller, Jul 08 2009

Keywords

• nonn

Comments

A number m is a term if and only if it has at least one non-infinitary divisor, or A000005(m) > A037445(m). - Vladimir Shevelev, Feb 23 2017

The asymptotic density of this sequence is 1 - A327839 = 0.3121728605... - Amiram Eldar, Jul 28 2020

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Complement of A036537.

A072587 is a subsequence.

Cf. A000005, A037445, A209229, A327839

Programs

Haskell

a162643 n = a162643_list !! (n-1)
a162643_list = filter ((== 0) . a209229 . a000005) [1..]
-- Reinhard Zumkeller, Nov 15 2012

Mathematica

Select[Range@ 192, ! IntegerQ@ Log2@ DivisorSigma[0, #] &] (* Michael De Vlieger, Feb 24 2017 *)

Python

from itertools import count, islice
from sympy import factorint
def A162643_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:any(map(lambda m:((k:=m+1)&-k)^k,factorint(n).values())),count(max(startvalue,1)))
A162643_list = list(islice(A162643_gen(),30)) # Chai Wah Wu, Jan 04 2023

Formula

A209229(A000005(a(n))) = 0. - Reinhard Zumkeller, Nov 15 2012