A123666 -id:A123666 - cp's OEIS frontend

A361201 Product of the right half (exclusive) of the multiset of prime factors of n; a(1) = 0.

0, 1, 1, 2, 1, 3, 1, 2, 3, 5, 1, 3, 1, 7, 5, 4, 1, 3, 1, 5, 7, 11, 1, 6, 5, 13, 3, 7, 1, 5, 1, 4, 11, 17, 7, 9, 1, 19, 13, 10, 1, 7, 1, 11, 5, 23, 1, 6, 7, 5, 17, 13, 1, 9, 11, 14, 19, 29, 1, 15, 1, 31, 7, 8, 13, 11, 1, 17, 23, 7, 1, 9, 1, 37, 5, 19, 11, 13, 1

Offset: 1

Gus Wiseman, Mar 10 2023

			The prime factors of 250 are {2,5,5,5}, with right half (exclusive) {5,5}, with product 25, so a(250) = 25.

Robert Israel, Table of n, a(n) for n = 1..10000

Positions of 1's are A000040.
Positions of first appearances are A123666.
The left inclusive version A347043.
The inclusive version is A347044.
The left version is A361200.
A000005 counts divisors.
A001221 counts distinct prime factors.
A006530 gives greatest prime factor.
A112798 lists prime indices, length A001222, sum A056239.
A360616 gives half of bigomega (exclusive), inclusive A360617.
A360673 counts multisets by right sum (exclusive), inclusive A360671.
First for prime indices, second for partitions, third for prime factors:
- A360676 gives left sum (exclusive), counted by A360672, product A361200.
- A360677 gives right sum (exclusive), counted by A360675, product A361201.
- A360678 gives left sum (inclusive), counted by A360675, product A347043.
- A360679 gives right sum (inclusive), counted by A360672, product A347044.
Cf. A001248, A026424, A096825, A347045, A347046, A360005.

f:= proc(n) local F;
  F:= ifactors(n)[2];
  F:= sort(map(t -> t[1]$t[2],F));
  convert(F[ceil(nops(F)/2)+1 ..-1],`*`)
end proc:
f(1):= 0:
map(f, [$1..100]); # Robert Israel, Aug 12 2024

Table[If[n==1,0,Times@@Take[Join@@ConstantArray@@@FactorInteger[n],-Floor[PrimeOmega[n]/2]]],{n,100}]

A361200(n) * A347044(n) = n.

A361201(n) * A347043(n) = n.

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

A123667 a(n) = n * 2^bigomega(n).

Original entry on oeis.org

1, 4, 6, 16, 10, 24, 14, 64, 36, 40, 22, 96, 26, 56, 60, 256, 34, 144, 38, 160, 84, 88, 46, 384, 100, 104, 216, 224, 58, 240, 62, 1024, 132, 136, 140, 576, 74, 152, 156, 640, 82, 336, 86, 352, 360, 184, 94, 1536, 196, 400, 204, 416, 106, 864, 220, 896, 228, 232, 118

Offset: 1

Franklin T. Adams-Watters, Oct 04 2006

Rearrangement of A123666.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A061142, A001222 (bigomega), A123666.

A123667:=n->n*2^numtheory[bigomega](n): seq(A123667(n), n=1..100); # Wesley Ivan Hurt, Apr 09 2017

Table[n*2^PrimeOmega[n],{n,60}] (* Harvey P. Dale, Jun 11 2014 *)

a(n) = n*2^bigomega(n); \\ Michel Marcus, Apr 10 2017

Totally multiplicative with a(p) = 2p, p prime.
a(n) = n * A061142(n).
Dirichlet g.f.: Product_{p prime} 1 / (1 - 2 * p^(1 - s)). - Ilya Gutkovskiy, Oct 29 2019

cp's OEIS Frontend

A361201 Product of the right half (exclusive) of the multiset of prime factors of n; a(1) = 0.

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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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A123667 a(n) = n * 2^bigomega(n).

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