A001918 Least positive primitive root of n-th prime. (Formerly M0242 N0083)

1, 2, 2, 3, 2, 2, 3, 2, 5, 2, 3, 2, 6, 3, 5, 2, 2, 2, 2, 7, 5, 3, 2, 3, 5, 2, 5, 2, 6, 3, 3, 2, 3, 2, 2, 6, 5, 2, 5, 2, 2, 2, 19, 5, 2, 3, 2, 3, 2, 6, 3, 7, 7, 6, 3, 5, 2, 6, 5, 3, 3, 2, 5, 17, 10, 2, 3, 10, 2, 2, 3, 7, 6, 2, 2, 5, 2, 5, 3, 21, 2, 2, 7, 5, 15, 2, 3, 13, 2, 3, 2, 13, 3, 2, 7, 5, 2, 3, 2, 2, 2, 2, 2, 3

Offset

1,2

Comments

If k is a primitive root of p=4m+1, then p-k is too. If k is a primitive root of p=4m+3, then p-k isn't, but has order 2m+1. - Jon Perry, Sep 07 2014

References

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 864.

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 213.

CRC Handbook of Combinatorial Designs, 1996, p. 615.

P. Fan and M. Darnell, Sequence Design for Communications Applications, Wiley, NY, 1996, Table A.1.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 111

Hua Loo Keng, Introduction To Number Theory, 'Table of least primitive roots for primes less than 50000', pp. 52-6, Springer NY 1982.

R. Osborn, Tables of All Primitive Roots of Odd Primes Less Than 1000, Univ. Texas Press, 1961.

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).

Links

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Loo-keng Hua, On the least primitive root of a prime, Bull. Amer. Math. Soc. 48 (1942), 726-730.

K. Matthews, Finding the least primitive root (mod p), p an odd prime

T. Oliveira e Silva, Least primitive root of prime numbers

Eric Weisstein's World of Mathematics, Primitive Root.

Example

modulo 7: 3^6=1, 3^2=2, 3^7=3, 3^4=4, 3^5=5, 3^3=6, 7=prime(4), 3=a(4).

Maple

A001918 := proc(n)
        numtheory[primroot](ithprime(n)) ;
end proc:

Mathematica

Table[PrimitiveRoot@Prime@n, {n, 101}] (* Robert G. Wilson v, Dec 15 2005 *)
PrimitiveRoot[Prime[Range[110]]] (* Harvey P. Dale, Jan 13 2013 *)

Prog

(PARI) for(x=1, 1000, print1(lift(znprimroot(prime(x))), ", "))
(Sage) [primitive_root(p) for p in primes(570)] # Zerinvary Lajos, May 24 2009
(Python)
from sympy import prime
from sympy.ntheory.residue_ntheory import primitive_root
def A001918(n): return primitive_root(prime(n)) # Chai Wah Wu, Sep 13 2022

Crossrefs

A column of A060749. Cf. A002233.

Sequence in context: A127808 A127809 A127810 * A268616 A331506 A002233

Adjacent sequences: A001915 A001916 A001917 * A001919 A001920 A001921

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Status

approved