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A345957
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Number of divisors of n with exactly half as many prime factors as n, counting multiplicity.
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22
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1, 0, 0, 1, 0, 2, 0, 0, 1, 2, 0, 0, 0, 2, 2, 1, 0, 0, 0, 0, 2, 2, 0, 2, 1, 2, 0, 0, 0, 0, 0, 0, 2, 2, 2, 3, 0, 2, 2, 2, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 2, 0, 0, 2, 2, 2, 2, 2, 0, 4, 0, 2, 0, 1, 2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 1, 2, 0, 4, 2, 2, 2
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OFFSET
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1,6
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COMMENTS
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These divisors do not necessarily include the central divisors (A207375), and may not themselves be central.
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LINKS
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EXAMPLE
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The a(n) divisors for selected n:
n = 1: 6: 36: 60: 210: 840: 900: 1260: 1296: 3600:
--------------------------------------------------------
1 2 4 4 6 8 12 12 16 16
3 6 6 10 12 18 18 24 24
9 10 14 20 20 20 36 36
15 15 28 30 28 54 40
21 30 45 30 81 60
35 42 50 42 90
70 75 45 100
105 63 150
70 225
105
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MATHEMATICA
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Table[Length[Select[Divisors[n], PrimeOmega[#]==PrimeOmega[n]/2&]], {n, 100}]
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PROG
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(PARI) a(n) = my(nb=bigomega(n)); sumdiv(n, d, 2*bigomega(d) == nb); \\ Michel Marcus, Aug 16 2021
(Python)
from sympy import divisors, factorint
def a(n):
npf = len(factorint(n, multiple=True))
divs = divisors(n)
return sum(2*len(factorint(d, multiple=True)) == npf for d in divs)
(Python 3.8+)
from itertools import combinations
from math import prod, comb
from sympy import factorint
if n == 1:
return 1
fs = factorint(n)
elist = list(fs.values())
q, r = divmod(sum(elist), 2)
k = len(elist)
if r:
return 0
c = 0
for i in range(k+1):
m = (-1)**i
for d in combinations(range(k), i):
t = k+q-sum(elist[j] for j in d)-i-1
if t >= 0:
c += m*comb(t, k-1)
(Python)
from sympy import factorint
from sympy.utilities.iterables import multiset_combinations
if n == 1:
return 1
fs = factorint(n, multiple=True)
q, r = divmod(len(fs), 2)
return 0 if r else len(list(multiset_combinations(fs, q))) # Chai Wah Wu, Aug 20 2021
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CROSSREFS
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The case of powers of 2 is A000035.
Positions of even terms are A000037.
Positions of odd terms are A000290.
The case of all divisors (not just 2) is A347042.
A001221 counts distinct prime factors.
A334997 counts chains of divisors of n by length.
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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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