A385552 Period of {binomial(N,n) mod 5: N in Z}.

1, 5, 5, 5, 5, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125, 125

Offset: 0

Author

Jianing Song, Jul 03 2025

Keywords

• nonn
• easy

Comments

a(n) is the smallest power of 5 > n. For the general result, see A349593.

Since the modulus (5) is a prime, the remainder of binomial(N,n) is given by Lucas's theorem.

Examples

			For N == 0, 1, ..., 24 (mod 5), binomial(N,5) == {0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4} (mod 5).
		

Links

• Jianing Song, Table of n, a(n) for n = 0..625
• Wikipedia, Lucas's theorem

Crossrefs

Column 5 of A349593. A062383, A064235 (if offset 0), A385553, and A385554 are respectively columns 2, 3, 6, and 10.

Programs

PARI

a(n) = if(n, 5^(logint(n,5)+1), 1)

Python

from sympy import integer_log
def A385552(n): return 5*5**(integer_log(n,5)[0]) if n else 1 # Chai Wah Wu, Jul 06 2025