A085311 Number of distinct 8th powers modulo n.

1, 2, 2, 2, 2, 4, 4, 2, 4, 4, 6, 4, 4, 8, 4, 2, 3, 8, 10, 4, 8, 12, 12, 4, 6, 8, 10, 8, 8, 8, 16, 2, 12, 6, 8, 8, 10, 20, 8, 4, 6, 16, 22, 12, 8, 24, 24, 4, 22, 12, 6, 8, 14, 20, 12, 8, 20, 16, 30, 8, 16, 32, 16, 3, 8, 24, 34, 6, 24, 16, 36, 8, 10, 20, 12, 20, 24, 16, 40, 4, 28, 12, 42, 16, 6

Offset: 1

Author

Labos Elemer, Jun 27 2003

Keywords

• nonn
• mult

Comments

This sequence is multiplicative. - Leon P Smith, Apr 16 2005

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• S. Li, On the number of elements with maximal order in the multiplicative group modulo n, Acta Arithm. 86 (2) (1998) 113, see proof of theorem 2.1
• R. J. Mathar, Size of the set of residues of integer powers of fixed exponent, (2017).

Crossrefs

Cf. A000224[k=2], A046530[k=3], A052273[k=4], A052274[k=5], A052275[k=6], A085310[k=7], A085312[k=9], A085313[k=10], A085314[k=11], A228849[k=12], A055653.

Programs

Maple

A085311 := proc(m)
    {seq( modp(b^8,m),b=0..m-1) };
    nops(%) ;
end proc:
seq(A085311(m),m=1..100) ; # R. J. Mathar, Sep 22 2017

Mathematica

a[n_] := Table[PowerMod[i, 8, n], {i, 0, n - 1}] // Union // Length;
Array[a, 100] (* Jean-François Alcover, Mar 24 2020 *)

PARI

a(n)=my(f=factor(n)); prod(i=1, #f[, 1], my(k=f[i, 1]^f[i, 2]); #vecsort(vector(k, i, i^8%k), , 8)) \\ Charles R Greathouse IV, Sep 05 2013