A196546 Numbers n such that the sum of the distinct residues of x^n (mod n), x=0..n-1, is divisible by n.

1, 3, 5, 7, 9, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 27, 28, 29, 30, 31, 33, 35, 37, 38, 39, 41, 43, 45, 46, 47, 49, 51, 52, 53, 55, 57, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 75, 77, 78, 79, 81, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 98, 99, 101

Offset: 1

Author

Michel Lagneau, Oct 03 2011

Keywords

• nonn

Comments

All odd prime numbers are in the sequence.

The sum of the distinct residues is 0, 1, 3, 1, 10, 8, 21, 1, 9, 25, 55, 14, 78, 42, 105, 1, 136,.. for n>=1.

Examples

			n= 14 is in the sequence because x^14 == 0, 1, 2, 4, 7, 8, 9, or 11 (mod 14), and the sum  0+1+2+4+7+8+9+11 = 42 is divisible by 14.
		

Crossrefs

Cf. A195637.

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
        if sumDistRes(n) mod n = 0 then
                printf("%d,",n);
        end if;
end do: # R. J. Mathar, Oct 04 2011

Mathematica

sumDistRes[n_] := Module[{re = {}, x}, For[x = 0, x <= n-1, x++, re = re ~Union~ {PowerMod[x, n, n]}]; Total[re]];
Select[Range[100], Mod[sumDistRes[#], #] == 0&] (* Jean-François Alcover, Oct 20 2023, after R. J. Mathar *)