A001918 Least positive primitive root of n-th prime.

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

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

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

Examples

			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).
		

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.
• Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 20.
• 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
• Jorma K. Merikoski, Pentti Haukkanen, and Timo Tossavainen, The congruence x^n = -a^n (mod m): Solvability and related OEIS sequences, Notes. Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 516-529. See p. 519.
• T. Oliveira e Silva, Least primitive root of prime numbers.
• Eric Weisstein's World of Mathematics, Primitive Root.

Crossrefs

A column of A060749. Cf. A002233.

Programs

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 *)

PARI

for(x=1, 1000, print1(lift(znprimroot(prime(x))), ", "))

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

Sage

[primitive_root(p) for p in primes(570)] # Zerinvary Lajos, May 24 2009