A385557 Period of {binomial(N,5) mod n: N in Z}. Also, period of {binomial(N,6) mod n: N in Z}.

1, 8, 9, 16, 25, 72, 7, 32, 27, 200, 11, 144, 13, 56, 225, 64, 17, 216, 19, 400, 63, 88, 23, 288, 125, 104, 81, 112, 29, 1800, 31, 128, 99, 136, 175, 432, 37, 152, 117, 800, 41, 504, 43, 176, 675, 184, 47, 576, 49, 1000, 153, 208, 53, 648, 275, 224, 171, 232, 59, 3600

Offset: 1

Author

Jianing Song, Jul 03 2025

Keywords

• nonn
• easy
• mult

Examples

			For N == 0, 1, ..., 71 (mod 72), binomial(N,5) == {0, 0, 0, 0, 0, 1, 0, 3, 2, 0, 0, 0, 0, 3, 4, 3, 0, 2, 0, 0, 0, 3, 0, 1, 0, 0, 2, 0, 0, 3, 0, 3, 4, 0, 0, 2, 0, 3, 0, 3, 0, 4, 0, 0, 2, 3, 0, 3, 0, 0, 4, 0, 0, 5, 0, 3, 0, 0, 0, 4, 0, 3, 2, 3, 0, 0, 0, 0, 4, 3, 0, 5} (mod 6), and binomial(N,6) == {0, 0, 0, 0, 0, 0, 1, 1, 4, 0, 0, 0, 0, 0, 3, 1, 4, 4, 0, 0, 0, 0, 3, 3, 4, 4, 4, 0, 0, 0, 3, 3, 0, 4, 4, 4, 0, 0, 3, 3, 0, 0, 4, 4, 4, 0, 3, 3, 0, 0, 0, 4, 4, 4, 3, 3, 0, 0, 0, 0, 4, 4, 1, 3, 0, 0, 0, 0, 0, 4, 1, 1} (mod 6).
		

Links

• Jianing Song, Table of n, a(n) for n = 1..10000

Crossrefs

Rows n = 5 and 6 of A349593. A022998, A385555, A385556, A385558, A385559, and A385560 are respectively rows 2, 3, 4, 7, 8, and 9-10.

Programs

Mathematica

A385557[n_] := If[n == 1, 1, n*Product[p^Floor[Log[p, 5]], {p, FactorInteger[n][[All, 1]]}]];
Array[A385557, 100] (* Paolo Xausa, Jul 07 2025 *)
a[n_] := n * GCD[n, 30] * (2 - Mod[n, 2]); Array[a, 100] (* Amiram Eldar, Jul 07 2025 *)

PARI

a(n, {choices=5}) = my(r=1, f=factor(n)); for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]); r *= p^(logint(choices, p)+e)); return(r)

Formula

Multiplicative with a(2^e) = 2^(e+2), a(3^e) = 3^(e+1), a(5^e) = 5^(e+1), and a(p^e) = p^e for primes p >= 7.

From Amiram Eldar, Jul 07 2025: (Start)

a(n) = n * gcd(30, n) * (2 - (n mod 2)).

Dirichlet g.f.: zeta(s-1) * (1 + 3/2*(s-1)) * (1 + 2/3*(s-1)) * (1 + 4/5*(s-1)).

Sum_{k=1..n} a(k) ~ (15/4) * n^2. (End)