

A002233


a(1) = 1; for n > 1, a(n) = least positive prime primitive root of nth 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
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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.  ZhiWei 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*(r1)/(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, SpringerVerlag, 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*(r1)/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*(r1)/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}] (* JeanFrançois Alcover, Mar 30 2011 *)
a[1] = 1; a[n_] := SelectFirst[PrimitiveRootList[Prime[n]], PrimeQ]; Array[a, 101] (* JeanFrançois 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: A001918 A268616 A331506 * A241516 A273458 A159953
Adjacent sequences: A002230 A002231 A002232 * A002234 A002235 A002236


KEYWORD

nonn,nice,easy


AUTHOR

N. J. A. Sloane


STATUS

approved



