A072587 Numbers having at least one prime factor with an even exponent.

4, 9, 12, 16, 18, 20, 25, 28, 36, 44, 45, 48, 49, 50, 52, 60, 63, 64, 68, 72, 75, 76, 80, 81, 84, 90, 92, 98, 99, 100, 108, 112, 116, 117, 121, 124, 126, 132, 140, 144, 147, 148, 150, 153, 156, 162, 164, 169, 171, 172, 175, 176, 180, 188, 192, 196, 198, 200, 204

Offset

1,1

Comments

Complement of the union of {1} and A002035. [Correction, Nov 15 2012]

A162645 is a subsequence and this sequence is a subsequence of A162643. - Reinhard Zumkeller, Jul 08 2009

The asymptotic density of this sequence is 1 - A065463 = 0.2955577990... - Amiram Eldar, Jul 21 2020

A number k is a term iff its core (A007913) properly divides its kernel (A007947), that is iff A336643(k) > 1. - David James Sycamore, Sep 18 2023

Links

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

Mathematica

Select[Range[210], MemberQ[EvenQ[Transpose[FactorInteger[#]][[2]]], True] &] (* Harvey P. Dale, Apr 03 2012 *)

Prog

(Haskell)
a072587 n = a072587_list !! (n-1)
a072587_list = tail $ filter (any even . a124010_row) [1..]
-- Reinhard Zumkeller, Nov 15 2012
(PARI) is(n)=n>3 && Set(factor(n)[, 2]%2)[1]==0 \\ Charles R Greathouse IV, Oct 16 2015
(Python)
from itertools import count, islice
from sympy import factorint
def A072587_gen(startvalue=1): # generator of terms
    return filter(lambda n:not all(map(lambda m:m&1, factorint(n).values())), count(max(startvalue, 1)))
A072587_list = list(islice(A072587_gen(), 30)) # Chai Wah Wu, Jan 04 2023

Crossrefs

Cf. A000037, A000203, A002035, A065463, A072588, A124010, A162643, A162645, A188999.

Cf. A007913, A007947, A336643.

Sequence in context: A355571 A072498 A162643 * A240112 A368714 A162645

Adjacent sequences: A072584 A072585 A072586 * A072588 A072589 A072590

Keyword

nonn

Author

Reinhard Zumkeller, Jun 23 2002

Extensions

Thanks to Zak Seidov, who noticed that 1 had to be removed. - Reinhard Zumkeller, Nov 15 2012

Status

approved