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

1,3

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)."

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.

Example

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.

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 *)

Crossrefs

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

Sequence in context: A067364 A090547 A087308 * A019759 A019965 A053148

Adjacent sequences: A121377 A121378 A121379 * A121381 A121382 A121383

Keyword

nice,nonn

Author

Ed Pegg Jr, Jul 25 2006

Status

approved