A056057 The largest square which divides n-th central binomial coefficient.

1, 1, 1, 1, 1, 4, 1, 1, 9, 36, 1, 4, 4, 4, 9, 9, 1, 4, 1, 4, 4, 4, 1, 4, 100, 100, 900, 900, 36, 144, 9, 9, 9, 36, 25, 100, 100, 100, 9, 36, 4, 4, 4, 4, 900, 3600, 225, 900, 1764, 1764, 1764, 1764, 196, 784, 4, 4, 4, 16, 4, 16, 16, 16, 441, 441, 49, 196, 49, 196, 36, 36, 1, 4

Offset

1,6

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = A008833(A001405(n)).

a(A046098(n)) = 1.

Mathematica

Table[First@ Select[Reverse@ Divisors@ Binomial[n, Floor[n/2]], IntegerQ@ Sqrt@ # &], {n, 72}] (* Michael De Vlieger, Feb 18 2017 *)
a[n_] := Times @@ (First[#]^(2*Floor[Last[#]/2]) & /@ FactorInteger[Binomial[n, Floor[n/2]]]); Array[a, 100] (* Amiram Eldar, Sep 06 2020 *)

Prog

(Python)
from math import prod
from sympy.ntheory.factor_ import digits
from sympy import primerange
def A056057(n):
    m = n>>1
    def s(n, p): return sum(digits(n, p)[1:])
    return prod(p**((s(m, p)+s(n-m, p)-s(n, p))//(p-1)&-2) for p in primerange(n+1)) # Chai Wah Wu, May 05 2026

Crossrefs

Cf. A001405, A000188, A008833, A007913, A055229, A055231, A046098, A034974.

Cf. A056056, A056058, A056059, A056060, A056061.

Sequence in context: A157108 A376553 A056647 * A226234 A369189 A185027

Adjacent sequences: A056054 A056055 A056056 * A056058 A056059 A056060

Keyword

nonn

Author

Labos Elemer, Jul 26 2000

Status

approved