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A002233 a(1) = 1; for n > 1, a(n) = least positive prime primitive root of n-th prime.
(Formerly M0243 N0084)
11

%I M0243 N0084

%S 1,2,2,3,2,2,3,2,5,2,3,2,7,3,5,2,2,2,2,7,5,3,2,3,5,2,5,2,11,3,3,2,3,2,

%T 2,7,5,2,5,2,2,2,19,5,2,3,2,3,2,7,3,7,7,11,3,5,2,43,5,3,3,2,5,17,17,2,

%U 3,19,2,2,3,7,11,2,2,5,2,5,3,29,2,2,7,5,17,2,3,13,2,3,2,13,3,2,7,5,2,3,2,2,2

%N a(1) = 1; for n > 1, a(n) = least positive prime primitive root of n-th prime.

%C According to Section F9 in Guy's book "Unsolved Problems in Number Theory" (Springer, 2004), P. Erdős asked whether for any large prime p there is a prime q < p so that q is a primitive root modulo p. See also the comments on A223942 related to this sequence. - _Zhi-Wei Sun_, Mar 29 2013

%C For n >= 2 the Dirichlet characters modulo prime(n), {Chi_{prime n}{(r,m)}, for n >= 1, r=1..(prime(n)-1) and m = 2..prime(n)-1, are determined from those for m = a(n), i.e., Chi_{prime n}(r,a(n)) = exp(2*Pi*I*(r-1)/(prime(n)-1)) and the power sequence S(n) := {a(n)^k (mod prime(n)), k = 1..(prime(n)-2)} by the strong multiplicity of Chi as Chi_{prime n}(r,m) = (Chi_{prime n}(r,a(n)))^{pos(m,S(n))} where S(n)_{pos(m,S(n))} = m. For m=1 Chi is always 1. For m = prime(n) Chi is always 0. For n=1 (prime 2) the characters are 1, 0 for r = 1 and m = 1, 2, respectively. See the example for a(4) below. - _Wolfdieter Lang_, Jan 19 2017

%D T. M. Apostol, An Introduction to Analytic Number Theory, Springer-Verlag, NY, 1976, 1986, p. 139.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D A. E. Western and J. C. P. Miller, Tables of Indices and Primitive Roots. Royal Society Mathematical Tables, Vol. 9, Cambridge Univ. Press, 1968, p. 2.

%H T. D. Noe, <a href="/A002233/b002233.txt">Table of n, a(n) for n = 1..10000</a>

%H A. E. Western and J. C. P. Miller, <a href="/A002223/a002223.pdf">Tables of Indices and Primitive Roots</a>, Royal Society Mathematical Tables, Vol. 9, Cambridge Univ. Press, 1968 [Annotated scans of selected pages]

%F a(n) = A122028(n) for n>1. - _Jonathan Sondow_, May 18 2017

%e n=4, a(4) = 3: Dirichlet characters for prime(4) = 7 from Chi_7(r,3) = exp(Pi*I*(r-1)/3) and the power sequence S(4) = [3, 2, 6, 4, 5]. Hence Chi_7(r,2) = Chi_7(r,3)^2 = exp(2*Pi*I*(r-1)/3), Chi_7(r,4) = Chi_7(r,3)^4, Chi_7(r,5) = Chi_7(r,3)^5, Chi_7(r,6) = Chi_7(r,3)^3. Chi_7(r,1) = 1 and Chi_7(r,7) = 0, for r=1..6. This produces the character modulo 7 table. See the Apostol reference, p. 139, with interchanged rows r = 2..6. - _Wolfdieter Lang_, Jan 19 2017

%t a[1] = 1; a[n_] := (p = Prime[n]; Select[Range[p], PrimeQ[#] && MultiplicativeOrder[#, p] == EulerPhi[p] &, 1]) // First; Table[a[n], {n, 100}] (* _Jean-François Alcover_, Mar 30 2011 *)

%t a[1] = 1; a[n_] := SelectFirst[PrimitiveRootList[Prime[n]], PrimeQ]; Array[a, 101] (* _Jean-François Alcover_, Sep 28 2016 *)

%o (PARI) leastroot(p)=forprime(q=2,p,if(znorder(Mod(q,p))+1==p,return(q)))

%o a(n)=if(n>1,leastroot(prime(n)),1) \\ _Charles R Greathouse IV_, Mar 20 2013

%Y See A122028 (least primitive root that is prime), A001918 (least positive primitive root), A223942.

%K nonn,nice,easy

%O 1,2

%A _N. J. A. Sloane_

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Last modified December 13 22:48 EST 2019. Contains 329974 sequences. (Running on oeis4.)