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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
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

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

REFERENCES

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

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

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

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.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

A. E. Western and J. C. P. Miller, Tables of Indices and Primitive Roots, Royal Society Mathematical Tables, Vol. 9, Cambridge Univ. Press, 1968 [Annotated scans of selected pages]

FORMULA

a(n) = A122028(n) for n>1. - Jonathan Sondow, May 18 2017

EXAMPLE

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

MATHEMATICA

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 *)

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

PROG

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

a(n)=if(n>1, leastroot(prime(n)), 1) \\ Charles R Greathouse IV, Mar 20 2013

CROSSREFS

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

Sequence in context: A127810 A001918 A268616 * A241516 A273458 A159953

Adjacent sequences:  A002230 A002231 A002232 * A002234 A002235 A002236

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified November 16 22:05 EST 2019. Contains 329208 sequences. (Running on oeis4.)