A255133 Number of unitary divisors of odd Catalan numbers (A038003).

1, 1, 2, 8, 64, 2048, 1048576, 68719476736, 18446744073709551616, 166153499473114484112975882535043072, 822752278660603021077484591278675252491367932816789931674304512, 4925250774549309901534880012517951725634967408808180833493536675530715221437151326426783281860614455100828498788352

Offset: 0

Author

Chai Wah Wu, Feb 15 2015

Keywords

• nonn

Examples

			A038003(4) = 9694845 which has 64 unitary divisors.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..14

Crossrefs

Cf. A038003, A034444, A119861, A119908, A120274, A120275, A255132.

Programs

Mathematica

a[n_] := 2^PrimeNu[CatalanNumber[2^n-1]]; Array[a, 12, 0] (* Amiram Eldar, Oct 04 2024 *)

PARI

a(n) = 1 << omega(binomial(2^(n+1)-2, 2^n-1)/(2^n)); \\ Amiram Eldar, Oct 04 2024

Python

from operator import mul
from functools import reduce
from sympy import factorint
A255133_list, c, s = [1, 1], {}, 3
for n in range(2, 2**16):
    for p, e in factorint(4*n-2).items():
        if p in c:
            c[p] += e
        else:
            c[p] = e
    for p, e in factorint(n+1).items():
        if c[p] == e:
            del c[p]
        else:
            c[p] -= e
    if n == s:
        c2 = 2**len(c)
        A255133_list.append(c2)
        s = 2*s+1

Formula

a(n) = A034444(A038003(n)).