%I #20 Aug 21 2021 13:22:04

%S 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,

%T 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,

%U 2,0,0,0,0,2,0,0,2,0,0,0,1,2,0,4,2,2,2

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

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

%e The a(n) divisors for selected n:

%e n = 1: 6: 36: 60: 210: 840: 900: 1260: 1296: 3600:

%e --------------------------------------------------------

%e 1 2 4 4 6 8 12 12 16 16

%e 3 6 6 10 12 18 18 24 24

%e 9 10 14 20 20 20 36 36

%e 15 15 28 30 28 54 40

%e 21 30 45 30 81 60

%e 35 42 50 42 90

%e 70 75 45 100

%e 105 63 150

%e 70 225

%e 105

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

%o (PARI) a(n) = my(nb=bigomega(n)); sumdiv(n, d, 2*bigomega(d) == nb); \\ _Michel Marcus_, Aug 16 2021

%o (Python)

%o from sympy import divisors, factorint

%o def a(n):

%o npf = len(factorint(n, multiple=True))

%o divs = divisors(n)

%o return sum(2*len(factorint(d, multiple=True)) == npf for d in divs)

%o print([a(n) for n in range(1, 88)]) # _Michael S. Branicky_, Aug 17 2021

%o (Python 3.8+)

%o from itertools import combinations

%o from math import prod, comb

%o from sympy import factorint

%o def A345957(n):

%o if n == 1:

%o return 1

%o fs = factorint(n)

%o elist = list(fs.values())

%o q, r = divmod(sum(elist),2)

%o k = len(elist)

%o if r:

%o return 0

%o c = 0

%o for i in range(k+1):

%o m = (-1)**i

%o for d in combinations(range(k),i):

%o t = k+q-sum(elist[j] for j in d)-i-1

%o if t >= 0:

%o c += m*comb(t,k-1)

%o return c # _Chai Wah Wu_, Aug 20 2021

%o (Python)

%o from sympy import factorint

%o from sympy.utilities.iterables import multiset_combinations

%o def A345957(n):

%o if n == 1:

%o return 1

%o fs = factorint(n,multiple=True)

%o q, r = divmod(len(fs),2)

%o return 0 if r else len(list(multiset_combinations(fs,q))) # _Chai Wah Wu_, Aug 20 2021

%Y The case of powers of 2 is A000035.

%Y Positions of even terms are A000037.

%Y Positions of odd terms are A000290.

%Y Positions of 0's are A026424.

%Y Positions of 1's are A056798.

%Y The rounded version is A096825.

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

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

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

%Y A000005 counts divisors.

%Y A001221 counts distinct prime factors.

%Y A001222 counts all prime factors.

%Y A056239 adds up prime indices, row sums of A112798.

%Y A207375 lists central divisors.

%Y A325534 counts separable partitions, ranked by A335433.

%Y A325535 counts inseparable partitions, ranked by A335448.

%Y A334997 counts chains of divisors of n by length.

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

%K nonn

%O 1,6

%A _Gus Wiseman_, Aug 16 2021