A058067 Number of polynomial functions from Z to Z/nZ.

1, 1, 4, 27, 64, 3125, 108, 823543, 1024, 19683, 12500, 285311670611, 1728, 302875106592253, 3294172, 84375, 65536, 827240261886336764177, 78732, 1978419655660313589123979, 200000, 22235661, 1141246682444, 20880467999847912034355032910567

Offset: 0

Author

N. J. A. Sloane, Nov 24 2000

Keywords

• nonn
• mult
• easy

Comments

The first formula for a(n) is due to Kempner (1921). - Jonathan Sondow, Nov 05 2017

Links

• Amiram Eldar, Table of n, a(n) for n = 0..388
• Manjul Bhargava, The factorial function and generalizations, Amer. Math. Monthly, Vol. 107, No. 9 (Nov. 2000), pp. 783-799; alternative link,
• Aubrey J. Kempner, Polynomials and their residue systems, continued, Amer. Math. Soc. Trans., Vol. 22 (1921), pp. 240-288.
• David Singmaster, On polynomial functions (mod m), Journal of Number Theory, Volume 6, Issue 5, October 1974, pp. 345-352.

Crossrefs

Cf. A051674, A090622, A240098.

Programs

Maple

A058067 := n->mul(n/gcd(n,k!),k=0..n-1);

Mathematica

a[0] = 1; a[n_] := Product[n/GCD[n, k!], {k, 0, n - 1}]; Array[a, 24, 0] (* Amiram Eldar, Sep 29 2020 *)

PARI

a(n) = prod(k=0, n-1, n/gcd(n, k!)); \\ Michel Marcus, Nov 06 2017

Formula

a(n) = Product_{k=0..n-1} n/gcd(n, k!).

Multiplicative with a(p^e) = p^t_p(e). - David W. Wilson, Aug 14 2005 [t_p(e) = Sum_{k>=0: e > A090622(k, p)} (e - A090622(k, p)) = p * Sum_{k = 1..e} max(0, k - A090622(e-k, p)). In particular, t_p(e) = p*e*(e+1)/2 for e <= p. - Andrey Zabolotskiy, Nov 09 2017 and Sep 29 2020]

a(prime(n)) = A051674(n). - R. J. Mathar, Apr 01 2014 [Edited by Andrey Zabolotskiy, Nov 08 2017]

a(n) = n^n / A240098(n). - Jonathan Sondow, Nov 10 2017