A162645 -id:A162645 - cp's OEIS frontend

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

Reinhard Zumkeller, Jun 23 2002

nonn

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

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

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

Cf. A007913, A007947, A336643.

a072587 n = a072587_list !! (n-1)
a072587_list = tail $ filter (any even . a124010_row) [1..]
-- Reinhard Zumkeller, Nov 15 2012

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

is(n)=n>3 && Set(factor(n)[,2]%2)[1]==0 \\ Charles R Greathouse IV, Oct 16 2015

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

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

A162511 Multiplicative function with a(p^e) = (-1)^(e-1).

Original entry on oeis.org

1, 1, 1, -1, 1, 1, 1, 1, -1, 1, 1, -1, 1, 1, 1, -1, 1, -1, 1, -1, 1, 1, 1, 1, -1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, 1, 1, -1, -1, -1, 1, -1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, -1, -1, 1, 1, 1, -1, 1, 1, 1, -1, 1, 1, -1, -1, 1, 1, 1, -1, -1, 1, 1, -1, 1, 1, 1, 1, 1, -1, 1, -1

Offset: 1

Gerard P. Michon, Jul 05 2009

G. C. Greubel, Table of n, a(n) for n = 1..5000
Gérard P. Michon, Multiplicative functions.

Cf. A005117, A076479, A162510, A162512, A002035, A072587, A036537, A162643, A162644, A162645, A046660, A008836, A307868.

A162511 := proc(n)
    local a,f;
    a := 1;
    for f in ifactors(n)[2] do
        a := a*(-1)^(op(2,f)-1) ;
    end do:
    return a;
end proc: # R. J. Mathar, May 20 2017

a[n_] := (-1)^(PrimeOmega[n] - PrimeNu[n]); Array[a, 100] (* Jean-François Alcover, Apr 24 2017, after Reinhard Zumkeller *)

a(n)=my(f=factor(n)[,2]); prod(i=1,#f,-(-1)^f[i]) \\ Charles R Greathouse IV, Mar 09 2015

from sympy import factorint
from operator import mul
def a(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [(-1)**(f[i] - 1) for i in f]) # Indranil Ghosh, May 20 2017

from functools import reduce
from sympy import factorint
def A162511(n): return -1 if reduce(lambda a,b:~(a^b), factorint(n).values(),0)&1 else 1 # Chai Wah Wu, Jan 01 2023

Multiplicative function with a(p^e)=(-1)^(e-1) for any prime p and any positive exponent e.
a(n) = 1 when n is a squarefree number (A005117).
From Reinhard Zumkeller, Jul 08 2009 (Start)
a(n) = (-1)^(A001222(n)-A001221(n)).
a(A162644(n)) = +1; a(A162645(n)) = -1. (End)
a(n) = A076479(n) * A008836(n). - R. J. Mathar, Mar 30 2011
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A307868. - Amiram Eldar, Sep 18 2022
Dirichlet g.f.: Product_{p prime} ((p^s + 2)/(p^s + 1)). - Amiram Eldar, Oct 26 2023

A162644 Numbers m such that A162511(m) = +1.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 93, 94, 95, 96, 97, 100

Offset: 1

Reinhard Zumkeller, Jul 08 2009

nonn

Also numbers n with A008836(n)=(-1)^A001221(n). - Enrique Pérez Herrero, Aug 03 2012

This sequence has an asymptotic density (1 + A065472/zeta(2))/2 = 0.735840... (Mossinghoff and Trudgian, 2019). - Amiram Eldar, Jul 07 2020

Amiram Eldar, Table of n, a(n) for n = 1..10000
Michael J. Mossinghoff and Timothy S. Trudgian, A tale of two omegas, arXiv:1906.02847 [math.NT], 2019.

Complement of A162645.
A002035 is a subsequence.
Cf. A008836, A001221, A036537, A065472.

Select[Range[100], EvenQ[PrimeOmega[#] - PrimeNu[#]] &] (* Amiram Eldar, Jul 07 2020 *)

A377816 Numbers that have a single even exponent in their prime factorization.

Original entry on oeis.org

4, 9, 12, 16, 18, 20, 25, 28, 44, 45, 48, 49, 50, 52, 60, 63, 64, 68, 72, 75, 76, 80, 81, 84, 90, 92, 98, 99, 108, 112, 116, 117, 121, 124, 126, 132, 140, 147, 148, 150, 153, 156, 162, 164, 169, 171, 172, 175, 176, 188, 192, 198, 200, 204, 207, 208, 212, 220, 228

Offset: 1

Amiram Eldar, Nov 09 2024

First differs from A162645 at n = 239: A162645(239) = 900 = 2^2 * 3^2 * 5^2 is not a term of this sequence.
Each term can be represented in a unique way as m * p^(2*k), k >= 1, where m is an exponentially odd number (A268335) and p is a prime that does not divide m.
Numbers k such that A350388(k) is a prime power with an even positive exponent (A056798 \ {1}).
The asymptotic density of this sequence is Product_{p prime} (1 - 1/(p*(p+1))) * Sum_{p prime} 1/(p^2+p-1) = 0.26256423811374124133... .

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

Cf. A056798, A065463, A162645, A229125 (odd analog), A268335, A350388, A377817.

A377818 is a subsequence.

Select[Range[250], Count[FactorInteger[#][[;; , 2]], _?EvenQ] == 1 &]

is(k) = if(k == 1, 0, my(e = factor(k)[, 2]); #select(x -> !(x%2), e) == 1);

A377817 Numbers that have more than one even exponent in their prime factorization.

Original entry on oeis.org

36, 100, 144, 180, 196, 225, 252, 300, 324, 396, 400, 441, 450, 468, 484, 576, 588, 612, 676, 684, 700, 720, 784, 828, 882, 900, 980, 1008, 1044, 1089, 1100, 1116, 1156, 1200, 1225, 1260, 1296, 1300, 1332, 1444, 1452, 1476, 1521, 1548, 1575, 1584, 1600, 1620, 1692, 1700, 1764, 1800

Offset: 1

Amiram Eldar, Nov 09 2024

Subsequence of A072413 and differs from it by not having the terms 216, 1000, 1080, 1512, ... .
Each term can be represented in a unique way as m * k^2, where m is an exponentially odd number (A268335) and k is a composite number that is coprime to m.
Numbers k such that A350388(k) is a square of a composite number (A062312 \ {1}).
The asymptotic density of this sequence is 1 - Product_{p prime} (1 - 1/(p*(p+1))) * (1 + Sum_{p prime} 1/(p^2+p-1)) = 0.032993560887093165933... .

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

Complement of the union of A268335 and A377816.
Subsequence of A072413.
Cf. A062312, A162645, A350388.

Select[Range[1800], Count[FactorInteger[#][[;; , 2]], _?EvenQ] > 1 &]

is(k) = if(k == 1, 0, my(e = factor(k)[, 2]); #select(x -> !(x%2), e) > 1);

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

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A162511 Multiplicative function with a(p^e) = (-1)^(e-1).

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A162644 Numbers m such that A162511(m) = +1.

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A377816 Numbers that have a single even exponent in their prime factorization.

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A377817 Numbers that have more than one even exponent in their prime factorization.

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