A347043 Smallest divisor of n with half (rounded up) as many prime factors (counting multiplicity) as n.

1, 2, 3, 2, 5, 2, 7, 4, 3, 2, 11, 4, 13, 2, 3, 4, 17, 6, 19, 4, 3, 2, 23, 4, 5, 2, 9, 4, 29, 6, 31, 8, 3, 2, 5, 4, 37, 2, 3, 4, 41, 6, 43, 4, 9, 2, 47, 8, 7, 10, 3, 4, 53, 6, 5, 4, 3, 2, 59, 4, 61, 2, 9, 8, 5, 6, 67, 4, 3, 10, 71, 8, 73, 2, 15, 4, 7, 6, 79, 8

Offset: 1

Author

Gus Wiseman, Aug 15 2021

Keywords

• nonn

Comments

Appears to contain every positive integer at least once.

This is correct. For any integer m, let p be any prime > m. Then a(m*p^A001222(m)) = m. - Sebastian Karlsson, Oct 11 2022

Examples

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

Links

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

Crossrefs

Positions of 2's are A001747.

Positions of odd terms are A005408.

Positions of even terms are A005843.

The case of powers of 2 is A016116.

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

These divisors are counted by A096825 (exact: A345957).

The greatest of these divisors is A347044 (exact: A347046).

The exact version is A347045.

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, A106529, A140271, A244991, A324522, A335433, A335448.

Programs

Mathematica

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

PARI

a(n) = my(bn=ceil(bigomega(n)/2)); fordiv(n, d, if (bigomega(d)==bn, return (d))); \\ Michel Marcus, Aug 18 2021

Python

from sympy import divisors, factorint
def a(n):
    npf = len(factorint(n, multiple=True))
    for d in divisors(n):
        if len(factorint(d, multiple=True)) == (npf+1)//2: return d
    return 1
print([a(n) for n in range(1, 81)]) # Michael S. Branicky, Aug 18 2021

Python

from math import prod
from sympy import factorint
def A347043(n):
    fs = factorint(n,multiple=True)
    l = len(fs)
    return prod(fs[:(l+1)//2]) # Chai Wah Wu, Aug 20 2021

Formula

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