A329812 Number of permutation polynomials (mod n).
1, 2, 6, 8, 120, 12, 5040, 128, 1296, 240, 39916800, 48, 6227020800, 10080, 720, 8192, 355687428096000, 2592, 121645100408832000, 960, 30240, 79833600, 25852016738884976640000, 768, 384000000, 12454041600, 25509168, 40320, 8841761993739701954543616000000, 1440
Offset: 1
Examples
For n=3, since it is a prime number, a(3) = 3! = 6. For n=4=2^2, a(4) = 2!*(2-1)^2*2^2 = 8.
Links
- Kenneth G. Hawes, Table of n, a(n) for n = 1..456
- Kenneth G. Hawes, SageMath program for generating the sequence
- Kenneth G. Hawes, Additional terms including those with more than 1000 digits, n = 1..5000
- G. Keller and F. R. Olson, Counting polynomial function (mod p^n), Duke Mathematical Journal, 35 (1968), 835-838.
Crossrefs
Formula involves the Kempner function A002034.
Formula
a(n) = Product_{i=1..r} a(p_i^k_i) for n having the unique prime factorization n = Product_{i=1..r} p_i^k_i.
a(p^k) = p! if k=1, a(p^k) = p!*(p-1)^p*p^p if k=2, and a(p^k) = p!*(p-1)^p*p^(p+f(p,k)) if k>2, where f(p,k) = Sum_{i=3..k} A002034(p^i).
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