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
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.
Programs
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Maple
A058067 := n->mul(n/gcd(n,k!),k=0..n-1);
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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(n) = n^n / A240098(n). - Jonathan Sondow, Nov 10 2017
Comments