A088145 Let p = prime(n); then a(n) = (Sum(primitive roots of p) - moebius(p-1))/p.

0, 1, 1, 1, 2, 2, 4, 3, 6, 6, 4, 6, 8, 6, 13, 12, 15, 8, 10, 15, 12, 14, 21, 20, 16, 20, 18, 27, 18, 24, 19, 27, 32, 24, 36, 22, 24, 28, 46, 42, 46, 24, 42, 32, 42, 35, 27, 34, 58, 36, 56, 53, 32, 52, 64, 71, 66, 39, 44, 48, 48, 72, 48, 66, 48, 78, 44, 48, 88, 56, 80

Offset: 1

Author

Ed Pegg Jr, Nov 03 2003

Keywords

• nonn

Comments

Gauss proved that the sum of the primitive roots of p is congruent to moebius(p-1) modulo p, for all primes p. - Jonathan Sondow, Feb 09 2013

Examples

			The primitive roots of prime(4) = 7 are 3 and 5, and moebius(7-1) = A008683(6) = 1, so a(4) = (3+5-1)/7 = 7/7 = 1. - _Jonathan Sondow_, Feb 10 2013
		

Links

• Wikipedia, Primitive root

Crossrefs

Cf. A008683, A060749, A123475, A222009.

Programs

Mathematica

PrimitiveRootQ[ a_Integer, p_Integer ] := Block[ {fac, res}, fac = FactorInteger[ p - 1 ]; res = Table[ PowerMod[ a, ( p - 1)/fac[ [ i, 1 ] ], p ], {i, Length[ fac ]} ]; ! MemberQ[ res, 1 ] ] PrimitiveRoots[ p_Integer ] := Select[ Range[ p - 1 ], PrimitiveRootQ[ #, p ] & ] Table[ (Total[ PrimitiveRoots[ Prime[ n ] ] ] - MoebiusMu[ Prime[ n ] - 1 ])/Prime[ n ], {n, 1, 100} ]
a[n_] := With[{p = Prime[n]}, Select[Range[p - 1], MultiplicativeOrder[#, p] == p - 1 &]]; Table[(Sum[a[n][[i]], {i, Length[a[n]]}] - MoebiusMu[Prime[n] - 1])/Prime[n], {n, 1,10}] (* Jonathan Sondow, Feb 09 2013 *)

Extensions

Definition corrected by Jonathan Sondow, Feb 09 2013