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A121380
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Sums of primitive roots for n (or 0 if n has no primitive roots).
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1
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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; internal format)
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OFFSET
| 1,3
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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)."
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REFERENCES
| J. C. F. Gauss, Disquisitiones Arithmeticae, 1801.
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LINKS
| T. D. Noe, Table of n, a(n) for n = 1..10000
Eric Weisstein, Primitive Roots.
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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.
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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}]
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CROSSREFS
| Cf. A001918, A008683, A046147 (primitive roots of n).
Sequence in context: A067364 A090547 A087308 * A019759 A019965 A053148
Adjacent sequences: A121377 A121378 A121379 * A121381 A121382 A121383
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KEYWORD
| nice,nonn
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AUTHOR
| Ed Pegg Jr (ed(AT)mathpuzzle.com), Jul 25 2006
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