A385559 Period of {binomial(N,8) mod n: N in Z}.

1, 16, 9, 32, 25, 144, 49, 64, 27, 400, 11, 288, 13, 784, 225, 128, 17, 432, 19, 800, 441, 176, 23, 576, 125, 208, 81, 1568, 29, 3600, 31, 256, 99, 272, 1225, 864, 37, 304, 117, 1600, 41, 7056, 43, 352, 675, 368, 47, 1152, 343, 2000, 153, 416, 53, 1296, 275, 3136, 171, 464, 59, 7200

Offset: 1

Author

Jianing Song, Jul 03 2025

Keywords

• nonn
• easy
• mult

Examples

			For N == 0, 1, ..., 63 (mod 64), binomial(N,8) == {0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 5, 5, 7, 7, 3, 3, 6, 6, 6, 6, 2, 2, 2, 2, 7, 7, 3, 3, 1, 1, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 1, 1, 3, 3, 7, 7, 2, 2, 2, 2, 6, 6, 6, 6, 3, 3, 7, 7, 5, 5, 1, 1} (mod 8).
		

Links

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

Crossrefs

Row n = 8 of A349593. A022998, A385555, A385556, A385557, A385558, and A385560 are respectively rows 2, 3, 4, 5-6, 7, and 9-10.

Programs

Mathematica

A385559[n_] := If[n == 1, 1, n*Product[p^Floor[Log[p, 8]], {p, FactorInteger[n][[All, 1]]}]];
Array[A385559, 100] (* Paolo Xausa, Jul 07 2025 *)
a[n_] := n * GCD[n, 210] * (4 - 3 * Mod[n, 2]); Array[a, 100] (* Amiram Eldar, Jul 07 2025 *)

PARI

a(n, {choices=8}) = 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+3), a(3^e) = 3^(e+1), a(5^e) = 5^(e+1), a(7^e) = 7^(e+1), and a(p^e) = p^e for primes p >= 11.

From Amiram Eldar, Jul 07 2025: (Start)

a(n) = n * gcd(210, n) * (4 - 3 * (n mod 2)).

Dirichlet g.f.: zeta(s-1) * (1 + 7/2*(s-1)) * (1 + 2/3*(s-1)) * (1 + 4/5*(s-1)) * (1 + 6/7*(s-1)).

Sum_{k=1..n} a(k) ~ (351/28) * n^2. (End)