A385555 Period of {binomial(N,3) mod n: N in Z}.

1, 4, 9, 8, 5, 36, 7, 16, 27, 20, 11, 72, 13, 28, 45, 32, 17, 108, 19, 40, 63, 44, 23, 144, 25, 52, 81, 56, 29, 180, 31, 64, 99, 68, 35, 216, 37, 76, 117, 80, 41, 252, 43, 88, 135, 92, 47, 288, 49, 100, 153, 104, 53, 324, 55, 112, 171, 116, 59, 360

Offset

1,2

Links

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

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,2,0,0,0,0,0,-1).

Formula

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

From Amiram Eldar, Jul 07 2025: (Start)

a(n) = n * gcd(6, n) = n * A089128(n).

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

Sum_{k=1..n} a(k) ~ (5/4) * n^2. (End)

G.f.: x*(1 + 4*x + 9*x^2 + 8*x^3 + 5*x^4 + 36*x^5 + 5*x^6 + 8*x^7 + 9*x^8 + 4*x^9 + x^10)/((1 - x)*(1 + x)*(1 - x + x^2)*(1 + x + x^2))^2. - Andrew Howroyd, Nov 12 2025

Example

For N == 0, 1, ..., 26 (mod 27), binomial(N,3) == {0, 0, 0, 1, 4, 1, 2, 8, 2, 3, 3, 3, 4, 7, 4, 5, 2, 5, 6, 6, 6, 7, 1, 7, 8, 5, 8} (mod 4).

Mathematica

A385555[n_] := If[n == 1, 1, n*Product[p^Floor[Log[p, 3]], {p, FactorInteger[n][[All, 1]]}]];
Array[A385555, 100] (* Paolo Xausa, Jul 07 2025 *)
a[n_] := n * GCD[n, 6]; Array[a, 100] (* Amiram Eldar, Jul 07 2025 *)

Prog

(PARI) a(n, {choices=3}) = 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)
(PARI) a(n) = n * gcd(6, n) \\ Andrew Howroyd, Nov 12 2025

Crossrefs

Row n = 3 of A349593. A022998, A385556, A385557, A385558, A385559, and A385560 are respectively rows n = 2, 4, 5-6, 7, 8, and 9-10.

Cf. A089128.

Sequence in context: A134902 A110992 A199203 * A370565 A370567 A387268

Adjacent sequences: A385552 A385553 A385554 * A385556 A385557 A385558

Keyword

nonn,easy,mult

Author

Jianing Song, Jul 03 2025

Status

approved