A379507 For n >= 1, a(n) = GCD((2*n)! / (n!)^2, 105).

1, 3, 5, 35, 21, 21, 3, 15, 5, 1, 21, 7, 35, 15, 15, 15, 15, 105, 105, 105, 15, 15, 15, 15, 21, 21, 7, 35, 105, 7, 7, 21, 105, 105, 21, 7, 7, 105, 35, 35, 105, 105, 105, 105, 105, 105, 105, 105, 15, 3, 3, 3, 105, 105, 21, 3, 3, 15, 15, 21, 21, 21, 15, 15, 15, 15

Offset

1,2

Comments

See the reference in the Links section, page 1747, Question. Is GCD(C(2*n, n), 105) = 1 for infinitely many positive integers n ? (is a(n) = 1 for infinitely many n >= 1 ?).

Links

Table of n, a(n) for n=1..66.

Daniel Berend and Jørgen E. Harmse, On polynomial-factorial diophantine equations, Transactions of the American Mathematical Society, Volume 358, Issue 4 (2006), pages 1741-1779.

Example

n = 3:  a(3) = GCD(6!/(3!)^2, 105) = 5.

Mathematica

a[n_] := GCD[Binomial[2*n, n], 105]; Array[a, 70] (* Amiram Eldar, Dec 23 2024 *)

Prog

(Python)
from math import prod
from sympy.ntheory.factor_ import digits
def A379507(n): return prod(p if (sum(digits(n, p)[1:])<<1)>sum(digits(n<<1, p)[1:]) else 1 for p in (3, 5, 7)) # Chai Wah Wu, May 07 2026

Crossrefs

Cf. A000142, A000984.

Sequence in context: A068111 A162444 A305858 * A261659 A346715 A259853

Adjacent sequences: A379504 A379505 A379506 * A379508 A379509 A379510

Keyword

nonn

Author

Ctibor O. Zizka, Dec 23 2024

Status

approved