A347045 Smallest divisor of n with exactly half as many prime factors (counting multiplicity) as n, or 1 if there are none.

1, 1, 1, 2, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 3, 4, 1, 1, 1, 1, 3, 2, 1, 4, 5, 2, 1, 1, 1, 1, 1, 1, 3, 2, 5, 4, 1, 2, 3, 4, 1, 1, 1, 1, 1, 2, 1, 1, 7, 1, 3, 1, 1, 6, 5, 4, 3, 2, 1, 4, 1, 2, 1, 8, 5, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 7, 1, 1, 1, 9, 2, 1, 4, 5, 2, 3

Offset

1,4

Links

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

Formula

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

Example

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

Mathematica

Table[If[#=={}, 1, Min[#]]&@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[[1 ;; np/2]]]]; Array[a, 100] (* Amiram Eldar, Nov 02 2024 *)

Prog

(Python)
from sympy import divisors, factorint
def a(n):
    npf = len(factorint(n, multiple=True))
    for d in divisors(n)[1:-1]:
        if 2*len(factorint(d, multiple=True)) == npf: return d
    return 1
print([a(n) for n in range(1, 88)]) # Michael S. Branicky, Aug 18 2021
(Python 3.8+)
from math import prod
from sympy import factorint
def A347045(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

Crossrefs

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

Positions of 1's are A026424.

Positions of even terms are A063745 = 2*A026424.

The case of powers of 2 is A072345.

Positions of 2's are A100484.

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

The rounded version is A347043.

The greatest divisor of this type is A347046 (rounded: A347044).

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, A001747, A027746, A038548, A056924, A063962, A106529, A140271, A244991, A324522, A335433, A335448.

Sequence in context: A350032 A379062 A146290 * A323345 A135539 A240060

Adjacent sequences: A347042 A347043 A347044 * A347046 A347047 A347048

Keyword

nonn

Author

Gus Wiseman, Aug 16 2021

Status

approved