%I
%S 1,2,5,8,23,26,68,57,139,174,123,222,328,257,612,636,886,488,669,1064,
%T 876,1105,1744,1780,1552,2020,1853,2890,1962,2712,2413,3536,4384,3335,
%U 5364,3322,3768,4564,7683,7266,8235,4344,8021,6176,8274
%N Sum of primitive roots of nth prime.
%C It is a result that goes back to Mirsky that the set of primes p for which p1 is squarefree has density A, where A denotes the Artin constant (A = prod_q (11/(q(q1)), q running over all primes). Numerically A = 0.3739558136.. = A005596. More precisely, Sum_{p <= x} mu(p1)^2 = Ax/log x + o(x/log x) as x tends to infinity. Conjecture: sum_{p <= x, mu(p1)=1} 1 = (A/2)x/log x + o(x/log x) and sum_{p <= x, mu(p1)=1} 1 = (A/2)x/log x + o(x/log x).  Pieter Moree (moree(AT)mpimbonn.mpg.de), Nov 03 2003
%C The number of the primitive roots is A008330(n).  _R. K. Guy_, Feb 25 2011
%C If prime(n) == 1 (mod 4), then a(n) = prime(n)*A008330(n)/2. There are also primes of the form prime(n) == 3 (mod 4) where prime(n)  a(n), namely prime(n) = 19, 127, 151, 163, 199, 251,... The list of primes in both modulo4 classes where prime(n)a(n) is 5, 13, 17, 19, 29, 37, 41, 53, 61,...  _R. K. Guy_, Feb 25 2011
%C a(n) = A076410(n) at n = 1, 3, 7, 55,... (for p = 2, 5, 17, 257... and perhaps only for the Fermat primes).  _R. K. Guy_, Feb 25 2011
%D C. F. Gauss, Disquisitiones Arithmeticae, Yale, 1965; see p. 52.
%H T. D. Noe, <a href="/A088144/b088144.txt">Table of n, a(n) for n=1..1000</a>
%H Leon Mirsky, <a href="http://www.jstor.org/stable/2305811">The Number of Representations of an Integer as the Sum of a Prime and a kFree Integer</a>, Amer. Math. Monthly 56 (1949), 1719.
%e For 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, the primitive roots are as follows: {{1}, {2}, {2, 3}, {3, 5}, {2, 6, 7, 8}, {2, 6, 7, 11}, {3, 5, 6, 7, 10, 11, 12, 14}, {2, 3, 10, 13, 14, 15}, {5, 7, 10, 11, 14, 15, 17, 19, 20, 21}, {2, 3, 8, 10, 11, 14, 15, 18, 19, 21, 26, 27}}
%t 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 ] & ]
%o (PARI) a(n)=local(r, p, pr, j); p=prime(n); r=vector(eulerphi(p1)); pr=znprimroot(p); for(i=1, p1, if(gcd(i, p1)==1, r[j++]=lift(pr^i))); vecsum(r) \\ after _Franklin T. AdamsWatters_'s code in A060749, _Michel Marcus_, Mar 16 2015
%Y Row sums of A060749, A254309.
%Y Cf. A088145, A121380, A123475, A222009.
%K nonn
%O 1,2
%A _Ed Pegg Jr_, Nov 03 2003
