A347044 Greatest divisor of n with half (rounded up) as many prime factors (counting multiplicity) as n.

1, 2, 3, 2, 5, 3, 7, 4, 3, 5, 11, 6, 13, 7, 5, 4, 17, 9, 19, 10, 7, 11, 23, 6, 5, 13, 9, 14, 29, 15, 31, 8, 11, 17, 7, 9, 37, 19, 13, 10, 41, 21, 43, 22, 15, 23, 47, 12, 7, 25, 17, 26, 53, 9, 11, 14, 19, 29, 59, 15, 61, 31, 21, 8, 13, 33, 67, 34, 23, 35, 71

Offset: 1

Author

Gus Wiseman, Aug 16 2021

Keywords

• nonn

Comments

Appears to contain each positive integer at least once, but only a finite number of times.

Examples

			The divisors of 123456 with half bigomega are: 16, 24, 5144, 7716, so a(123456) = 7716.
		

Links

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

Crossrefs

The greatest divisor without the condition is A006530 (smallest: A020639).

Divisors of this type are counted by A096825 (exact: A345957).

The case of powers of 2 is A163403.

The smallest divisor of this type is given by A347043 (exact: A347045).

The exact version is A347046.

A000005 counts divisors.

A001221 counts distinct prime factors.

A001222 counts all prime factors (also called bigomega).

A038548 counts inferior (or superior) divisors (strict: A056924).

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, A060775, A106529, A217581, A244990, A335433, A335448.

Programs

Mathematica

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

Python

from sympy import divisors, factorint
def a(n):
    npf = len(factorint(n, multiple=True))
    for d in divisors(n)[::-1]:
        if len(factorint(d, multiple=True)) == (npf+1)//2: return d
    return 1
print([a(n) for n in range(1, 72)]) # Michael S. Branicky, Aug 18 2021

Python

from math import prod
from sympy import factorint
def A347044(n):
    fs = factorint(n,multiple=True)
    l = len(fs)
    return prod(fs[l//2:]) # Chai Wah Wu, Aug 20 2021

Formula

a(n) = Product_{k=floor(A001222(n)/2)+1..A001222(n)} A027746(n,k). - Amiram Eldar, Nov 02 2024