A195812 Sum of the distinct residues of x^n (mod n), x=0..n-1.

0, 1, 3, 1, 10, 8, 21, 1, 9, 25, 55, 14, 78, 42, 105, 1, 136, 20, 171, 22, 84, 110, 253, 26, 50, 169, 27, 84, 406, 150, 465, 1, 528, 289, 595, 38, 666, 342, 273, 42, 820, 130, 903, 198, 315, 460, 1081, 50, 147, 125, 1275, 156, 1378, 56, 385, 140, 570, 841

Offset: 1

Author

Michel Lagneau, Oct 05 2011

Keywords

• nonn

Comments

a(n) has the following properties :

If n is a power of 2 then a(n)= 1 ;

Except for n = 9 where a(9)=9, if a(n) is the square of a prime p, the sequence shows that n is of the form n = 2p.

The numbers m such that a(m) are square are : 1, 2, 4, 8, 9, 10, 16, 26, 32, 34, 58, 64, 74, 81, ...

Examples

			a(10) = 25 because the residues (mod 10) of x^10 are 0, 1, 4, 5, 6, 9 and the sum 25 is a square => a(10) = a(2*5)= 5^2.
		

Links

• Zak Seidov, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A196546, A196547.

Programs

Maple

sumDistRes := proc(n)
        local re, x, r ;
        re := {} ;
        for x from 0 to n-1 do
                re := re union { modp(x^n, n) } ;
        end do:
        add(r, r=re) ;
end proc:
for n from 1 to 100 do
          printf("%d, ", sumDistRes(n));
end do: # (Program of R. J. Mathar - see A196546)

Mathematica

Table[{m,Total[Union[Table[PowerMod[x,m,m],{x,m-1}]]]},{m,1000}] (* Zak Seidov, Oct 06 2011 *)

PARI

a(n) = vecsum(Set(vector(n, k, lift(Mod(k-1,n)^n)))); \\ Michel Marcus, Jun 01 2015