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A347045
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Smallest divisor of n with exactly half as many prime factors (counting multiplicity) as n, or 1 if there are none.
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8
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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
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OFFSET
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1,4
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LINKS
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EXAMPLE
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The divisors of 90 with half bigomega are: 6, 9, 10, 15, so a(90) = 6.
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MATHEMATICA
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Table[If[#=={}, 1, Min[#]]&@Select[Divisors[n], PrimeOmega[#]==PrimeOmega[n]/2&], {n, 100}]
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PROG
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(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
(Python 3.8+)
from math import prod
from sympy import factorint
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
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CROSSREFS
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The smallest divisor without the condition is A020639 (greatest: A006530).
The case of powers of 2 is A072345.
A001221 counts distinct prime factors.
A001222 counts all prime factors (also called bigomega).
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, A038548, A056924, A063962, A106529, A140271, A244991, A324522, A335433, A335448.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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