A056061 Number of square divisors of central binomial coefficients.

1, 1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 4, 4, 8, 8, 4, 6, 2, 2, 2, 4, 2, 4, 4, 4, 2, 4, 2, 2, 2, 2, 8, 12, 4, 8, 8, 8, 8, 8, 4, 6, 2, 2, 2, 3, 2, 3, 3, 3, 4, 4, 2, 4, 2, 4, 4, 4, 1, 2, 2, 2, 4, 4, 8, 12, 2, 4, 12, 12, 4, 4, 8, 12, 12, 12, 4, 6, 8, 12, 12, 12, 8, 16, 8, 8, 6

Offset

1,6

Links

Michel Marcus, Table of n, a(n) for n = 1..120

Formula

a(n) = A046951(A001405(n)) = A000005(A000188(A001405(n))).

Example

n=27: binomial(27,13) = 20058300, its largest square-divisor is 900=30^2 so a(27) = tau(30) = 8.

Mathematica

Table[Count[Divisors@ Binomial[n, Floor[n/2]], d_ /; IntegerQ@ Sqrt@ d], {n, 0, 84}] (* Michael De Vlieger, Feb 18 2017 *)

Prog

(PARI) a(n) = sumdiv(binomial(n, n\2), d, issquare(d)); \\ Michel Marcus, Feb 19 2017
(Python)
from math import prod
from sympy import primerange
from sympy.ntheory.factor_ import digits
def A056061(n):
    m = n>>1
    def s(n, p): return sum(digits(n, p)[1:])
    return prod(((s(m, p)+s(n-m, p)-s(n, p))//(p-1)>>1)+1 for p in primerange(n+1)) # Chai Wah Wu, May 05 2026

Crossrefs

Cf. A001405, A046951, A000005, A000188, A056056, A056057, A056058, A056059, A056060.

Sequence in context: A389709 A376555 A056648 * A029265 A103648 A293176

Adjacent sequences: A056058 A056059 A056060 * A056062 A056063 A056064

Keyword

nonn

Author

Labos Elemer, Jul 26 2000

Status

approved