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

1,2

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.

Links

Table of n, a(n) for n=1..66.

Example

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.

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 *)

Crossrefs

Cf. A195637.

Sequence in context: A081534 A214547 A097218 * A371179 A231773 A007617

Adjacent sequences: A196543 A196544 A196545 * A196547 A196548 A196549

Keyword

nonn

Author

Michel Lagneau, Oct 03 2011

Status

approved