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 A121380 Sums of primitive roots for n (or 0 if n has no primitive roots). 2
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified August 3 20:08 EDT 2020. Contains 336201 sequences. (Running on oeis4.)