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

%I A385552 #17 Jul 06 2025 12:22:53
%S A385552 1,5,5,5,5,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,
%T A385552 25,125,125,125,125,125,125,125,125,125,125,125,125,125,125,125,125,
%U A385552 125,125,125,125,125,125,125,125,125,125,125,125,125,125,125,125,125,125,125,125
%N A385552 Period of {binomial(N,n) mod 5: N in Z}.
%C A385552 a(n) is the smallest power of 5 > n. For the general result, see A349593.
%C A385552 Since the modulus (5) is a prime, the remainder of binomial(N,n) is given by Lucas's theorem.
%H A385552 Jianing Song, <a href="/A385552/b385552.txt">Table of n, a(n) for n = 0..625</a>
%H A385552 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lucas%27s_theorem">Lucas's theorem</a>
%e A385552 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).
%o A385552 (PARI) a(n) = if(n, 5^(logint(n,5)+1), 1)
%o A385552 (Python)
%o A385552 from sympy import integer_log
%o A385552 def A385552(n): return 5*5**(integer_log(n,5)[0]) if n else 1 # _Chai Wah Wu_, Jul 06 2025
%Y A385552 Column 5 of A349593. A062383, A064235 (if offset 0), A385553, and A385554 are respectively columns 2, 3, 6, and 10.
%K A385552 nonn,easy
%O A385552 0,2
%A A385552 _Jianing Song_, Jul 03 2025