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A347044
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Greatest divisor of n with half (rounded up) as many prime factors (counting multiplicity) as n.
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22
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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
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OFFSET
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1,2
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COMMENTS
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Appears to contain each positive integer at least once, but only a finite number of times.
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LINKS
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EXAMPLE
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The divisors of 123456 with half bigomega are: 16, 24, 5144, 7716, so a(123456) = 7716.
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MATHEMATICA
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Table[Max[Select[Divisors[n], PrimeOmega[#]==Ceiling[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]:
if len(factorint(d, multiple=True)) == (npf+1)//2: return d
return 1
(Python 3.8+)
from math import prod
from sympy import factorint
fs = factorint(n, multiple=True)
l = len(fs)
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CROSSREFS
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The greatest divisor without the condition is A006530 (smallest: A020639).
The case of powers of 2 is A163403.
The smallest divisor of this type is given by A347043 (exact: A347045).
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).
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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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