A345957 Number of divisors of n with exactly half as many prime factors as n, counting multiplicity.

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

Offset: 1

Author

Gus Wiseman, Aug 16 2021

Keywords

• nonn

Comments

These divisors do not necessarily include the central divisors (A207375), and may not themselves be central.

Examples

			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
		

Crossrefs

The case of powers of 2 is A000035.

Positions of even terms are A000037.

Positions of odd terms are A000290.

Positions of 0's are A026424.

Positions of 1's are A056798.

The rounded version is A096825.

The case of all divisors (not just 2) is A347042.

The smallest of these divisors is A347045 (rounded: A347043).

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

A000005 counts divisors.

A001221 counts distinct prime factors.

A001222 counts all prime factors.

A056239 adds up prime indices, row sums of A112798.

A207375 lists central divisors.

A325534 counts separable partitions, ranked by A335433.

A325535 counts inseparable partitions, ranked by A335448.

A334997 counts chains of divisors of n by length.

Cf. A001227, A001414, A028260, A033676, A033677, A073093, A074206, A217581, A344653, A346697-A346704.

Programs

Mathematica

Table[Length[Select[Divisors[n],PrimeOmega[#]==PrimeOmega[n]/2&]],{n,100}]

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)
print([a(n) for n in range(1, 88)]) # Michael S. Branicky, Aug 17 2021
(Python 3.8+)
from itertools import combinations
from math import prod, comb
from sympy import factorint
def A345957(n):
    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)
    return c # Chai Wah Wu, Aug 20 2021

Python

from sympy import factorint
from sympy.utilities.iterables import multiset_combinations
def A345957(n):
    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