A197929 - OEIS

A197929

Number of distinct residues of x^(n-1) (mod n), x=0..n-1.

1, 2, 2, 3, 2, 6, 2, 5, 4, 10, 2, 9, 2, 14, 6, 9, 2, 14, 2, 15, 8, 22, 2, 15, 6, 26, 10, 9, 2, 30, 2, 17, 12, 34, 12, 21, 2, 38, 14, 25, 2, 42, 2, 33, 8, 46, 2, 27, 8, 42, 18, 15, 2, 38, 18, 35, 20, 58, 2, 45, 2, 62, 16, 33, 8, 18, 2, 51, 24, 30, 2, 35, 2, 74

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OFFSET

1,2

COMMENTS

a(n) = 2 if n prime because the residues are 0 and 1 (Fermat's little theorem).

a(n) = n if n = 2p, p prime > 2. But there exists nonprime numbers q such that a(2q) = 2q, for example q = 1, 15, 21, 39,...

LINKS

Table of n, a(n) for n=1..74.

EXAMPLE

a(8) = 5 because x^7 == 0, 1, 3, 5, 7 (mod 8) => 5 distinct residues.

MATHEMATICA

Length[Union[#]]& /@ Table[Mod[k^(n-1), n], {n, 74}, {k, n}]