A123667 -id:A123667 - cp's OEIS frontend

A061142 Replace each prime factor of n with 2: a(n) = 2^bigomega(n), where bigomega = A001222, number of prime factors counted with multiplicity.

1, 2, 2, 4, 2, 4, 2, 8, 4, 4, 2, 8, 2, 4, 4, 16, 2, 8, 2, 8, 4, 4, 2, 16, 4, 4, 8, 8, 2, 8, 2, 32, 4, 4, 4, 16, 2, 4, 4, 16, 2, 8, 2, 8, 8, 4, 2, 32, 4, 8, 4, 8, 2, 16, 4, 16, 4, 4, 2, 16, 2, 4, 8, 64, 4, 8, 2, 8, 4, 8, 2, 32, 2, 4, 8, 8, 4, 8, 2, 32, 16, 4, 2, 16, 4, 4, 4, 16, 2, 16, 4, 8, 4, 4, 4

Offset: 1

Henry Bottomley, May 29 2001

The inverse Möbius transform of A162510. - R. J. Mathar, Feb 09 2011

			a(100)=16 since 100=2*2*5*5 and so a(100)=2*2*2*2.

R. Zumkeller, Table of n, a(n) for n = 1..10005
R. J. Mathar, Survey of Dirichlet Series of Multiplicative Arithmetic Functions, arXiv:1106.4038 [math.NT], 2011-2012. See eq. (2.12).
Index entries for sequences computed from exponents in factorization of n

Cf. A000079, A001222, A001316, A034444, A069205 (partial sums), A123667, A124508, A156552.

with(numtheory): seq(2^bigomega(n),n=1..95);

Table[2^PrimeOmega[n], {n, 1, 95}] (* Jean-François Alcover, Jun 08 2013 *)

a(n)=direuler(p=1,n,1/(1-2*X))[n] /* Ralf Stephan, Mar 28 2015 */

a(n) = 2^bigomega(n); \\ Michel Marcus, Aug 08 2017

a(n) = Sum_{d divides n} 2^(bigomega(d)-omega(d)) = Sum_{d divides n} 2^(A001222(d) - A001221(d)). - Benoit Cloitre, Apr 30 2002
a(n) = A000079(A001222(n)), i.e., a(n)=2^bigomega(n). - Emeric Deutsch, Feb 13 2005
Totally multiplicative with a(p) = 2. - Franklin T. Adams-Watters, Oct 04 2006
Dirichlet g.f.: Product_{p prime} 1/(1-2*p^(-s)). - Ralf Stephan, Mar 28 2015
a(n) = A001316(A156552(n)). - Antti Karttunen, May 29 2017
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} 1/(1 - 1/(p^s - 1)^2). - Vaclav Kotesovec, Mar 14 2023

A347046 Greatest divisor of n with exactly half as many prime factors (counting multiplicity) as n, or 1 if there are none.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 1, 3, 5, 1, 1, 1, 7, 5, 4, 1, 1, 1, 1, 7, 11, 1, 6, 5, 13, 1, 1, 1, 1, 1, 1, 11, 17, 7, 9, 1, 19, 13, 10, 1, 1, 1, 1, 1, 23, 1, 1, 7, 1, 17, 1, 1, 9, 11, 14, 19, 29, 1, 15, 1, 31, 1, 8, 13, 1, 1, 1, 23, 1, 1, 1, 1, 37, 1, 1, 11, 1, 1, 1, 9

Offset: 1

Gus Wiseman, Aug 16 2021

nonn

Problem: What are the positions of last appearances > 1?

			The divisors of 90 with half bigomega are: 6, 9, 10, 15, so a(90) = 15.

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

The greatest divisor without the condition is A006530 (smallest: A020639).
Positions of 1's are A026424.
The case of powers of 2 is A072345.
Positions of first appearances are A123667 (sorted: A123666).
Divisors of this type are counted by A345957 (rounded: A096825).
The rounded version is A347044.
The smallest divisor of this is A347045 (rounded: A347043).
A000005 counts divisors.
A001221 counts distinct prime factors.
A001222 counts all prime factors (also called bigomega).
A056239 adds up prime indices, row sums of A112798.
A207375 lists central divisors (min: A033676, max: A033677).
A340387 lists numbers whose sum of prime indices is twice bigomega.
A340609 lists numbers whose maximum prime index divides bigomega.
A340610 lists numbers whose maximum prime index is divisible by bigomega.
A347042 counts divisors d|n such that bigomega(d) divides bigomega(n).
Cf. A000720, A027746, A038548, A060775, A106529, A244991, A335433, A335448.

Table[If[#=={},1,Max[#]]&@Select[Divisors[n], PrimeOmega[#]==PrimeOmega[n]/2&],{n,100}]
a[n_] := Module[{p = Flatten[Table[#[[1]], {#[[2]]}] & /@ FactorInteger[n]], np}, np = Length[p]; If[OddQ[np], 1, Times @@ p[[np/2+1 ;; np]]]]; Array[a, 100] (* Amiram Eldar, Nov 02 2024 *)

from sympy import divisors, factorint
def a(n):
    npf = len(factorint(n, multiple=True))
    for d in divisors(n)[-1:0:-1]:
        if 2*len(factorint(d, multiple=True)) == npf: return d
    return 1
print([a(n) for n in range(1, 82)]) # Michael S. Branicky, Aug 18 2021

from math import prod
from sympy import factorint
def A347046(n):
    fs = factorint(n,multiple=True)
    q, r = divmod(len(fs),2)
    return 1 if r else prod(fs[q:]) # Chai Wah Wu, Aug 20 2021

a(n) = Product_{k=A001222(n)/2+1..A001222(n)} A027746(n,k) if A001222(n) is even, and 1 otherwise. - Amiram Eldar, Nov 02 2024

A123666 Numbers with an even number of prime factors, at least half of which are 2.

Original entry on oeis.org

1, 4, 6, 10, 14, 16, 22, 24, 26, 34, 36, 38, 40, 46, 56, 58, 60, 62, 64, 74, 82, 84, 86, 88, 94, 96, 100, 104, 106, 118, 122, 132, 134, 136, 140, 142, 144, 146, 152, 156, 158, 160, 166, 178, 184, 194, 196, 202, 204, 206, 214, 216, 218, 220, 224, 226, 228, 232, 240

Offset: 1

Franklin T. Adams-Watters, Oct 04 2006

nonn

This is the lexicographically earliest maximal set of nonprime integers that has unique factorization. Products of any number of terms from A100484, with repetition allowed.

James C. McMahon, Table of n, a(n) for n = 1..10000

Cf. A100484, A002808, A123667.

fn[b_]:=Total[Last/@FactorInteger[b]];Join[{1},Select[Range[240],EvenQ[#]&&EvenQ[fn[#]]&&First[FactorInteger[#]][[2]]>=fn[#]/2&]] (* James C. McMahon, Nov 23 2024 *)

A328854 Dirichlet g.f.: Product_{p prime} 1 / (1 - 2 * p^(-s))^2.

Original entry on oeis.org

1, 4, 4, 12, 4, 16, 4, 32, 12, 16, 4, 48, 4, 16, 16, 80, 4, 48, 4, 48, 16, 16, 4, 128, 12, 16, 32, 48, 4, 64, 4, 192, 16, 16, 16, 144, 4, 16, 16, 128, 4, 64, 4, 48, 48, 16, 4, 320, 12, 48, 16, 48, 4, 128, 16, 128, 16, 16, 4, 192, 4, 16, 48, 448, 16, 64, 4, 48, 16, 64, 4, 384, 4, 16, 48

Offset: 1

Ilya Gutkovskiy, Oct 28 2019

Dirichlet convolution of A061142 with itself.

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

Cf. A000005, A001222, A061142, A123667, A322328.

Table[2^PrimeOmega[n] DivisorSigma[0, n], {n, 1, 75}]
f[p_, e_] := (e + 1)*2^e; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 02 2020 *)

a(n) = numdiv(n)*2^bigomega(n); \\ Michel Marcus, Dec 02 2020

for(n=1, 100, print1(direuler(p=2, n, 1/(1 - 2*X)^2)[n], ", ")) \\ Vaclav Kotesovec, Aug 22 2021

If n = Product (p_j^k_j) then a(n) = Product (2^k_j * (k_j + 1)).

a(n) = 2^bigomega(n) * tau(n), where bigomega = A001222 and tau = A000005.

cp's OEIS Frontend

A061142 Replace each prime factor of n with 2: a(n) = 2^bigomega(n), where bigomega = A001222, number of prime factors counted with multiplicity.

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A347046 Greatest divisor of n with exactly half as many prime factors (counting multiplicity) as n, or 1 if there are none.

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A123666 Numbers with an even number of prime factors, at least half of which are 2.

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A328854 Dirichlet g.f.: Product_{p prime} 1 / (1 - 2 * p^(-s))^2.

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Mathematica

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