

A088144


Sum of primitive roots of nth prime.


13



1, 2, 5, 8, 23, 26, 68, 57, 139, 174, 123, 222, 328, 257, 612, 636, 886, 488, 669, 1064, 876, 1105, 1744, 1780, 1552, 2020, 1853, 2890, 1962, 2712, 2413, 3536, 4384, 3335, 5364, 3322, 3768, 4564, 7683, 7266, 8235, 4344, 8021, 6176, 8274
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OFFSET

1,2


COMMENTS

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
The number of the primitive roots is A008330(n).  R. K. Guy, Feb 25 2011
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
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


REFERENCES

C. F. Gauss, Disquisitiones Arithmeticae, Yale, 1965; see p. 52.


LINKS

T. D. Noe, Table of n, a(n) for n=1..1000
Leon Mirsky, The Number of Representations of an Integer as the Sum of a Prime and a kFree Integer, Amer. Math. Monthly 56 (1949), 1719.


EXAMPLE

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}}


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 ] & ]


PROG

(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


CROSSREFS

Row sums of A060749, A254309.
Cf. A088145, A121380, A123475, A222009.
Sequence in context: A137095 A092097 A195295 * A100501 A291605 A142869
Adjacent sequences: A088141 A088142 A088143 * A088145 A088146 A088147


KEYWORD

nonn


AUTHOR

Ed Pegg Jr, Nov 03 2003


STATUS

approved



