A121380 Sums of primitive roots for n (or 0 if n has no primitive roots).

0, 1, 2, 3, 5, 5, 8, 0, 7, 10, 23, 0, 26, 8, 0, 0, 68, 16, 57, 0, 0, 56, 139, 0, 100, 52, 75, 0, 174, 0, 123, 0, 0, 136, 0, 0, 222, 114, 0, 0, 328, 0, 257, 0, 0, 208, 612, 0, 300, 200, 0, 0, 636, 156, 0, 0, 0, 348, 886, 0, 488, 216, 0, 0, 0, 0, 669, 0, 0, 0

Offset: 1

Author

Ed Pegg Jr, Jul 25 2006

Keywords

• nice
• nonn

Comments

In Article 81 of his Disquisitiones Arithmeticae (1801), Gauss proves that the sum of all primitive roots (A001918) of a prime p, mod p, equals MoebiusMu[p-1] (A008683). "The sum of all primitive roots is either = 0 (mod p) (when p-1 is divisible by a square), or = +-1 (mod p) (when p-1 is the product of unequal prime numbers; if the number of these is even the sign is positive but if the number is odd, the sign is negative)."

Examples

			The primitive roots of 13 are 2, 6, 7, 11. Their sum is 26, or 0 (mod 13). By Gauss, 13-1=12 is thus divisible by a square number.
		

References

• J. C. F. Gauss, Disquisitiones Arithmeticae, 1801.

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Eric Weisstein, Primitive Roots.

Crossrefs

Cf. A001918, A008683, A046147 (primitive roots of n), A088144, A088145, A123475, A222009.

Programs

Mathematica

primitiveRoots[n_] := If[n == 1, {}, If[n == 2, {1}, Select[Range[2, n-1], MultiplicativeOrder[#, n] == EulerPhi[n] &]]]; Table[Total[primitiveRoots[n]], {n,100}]
(* From version 10 up: *)
Table[Total @ PrimitiveRootList[n], {n, 1, 100}] (* Jean-François Alcover, Oct 31 2016 *)