A023048 Smallest prime having least positive primitive root n, or 0 if no such prime exists.

2, 3, 7, 0, 23, 41, 71, 0, 0, 313, 643, 4111, 457, 1031, 439, 0, 311, 53173, 191, 107227, 409, 3361, 2161, 533821, 0, 12391, 0, 133321, 15791, 124153, 5881, 0, 268969, 48889, 64609, 0, 36721, 55441, 166031, 1373989, 156601, 2494381, 95471, 71761, 95525767

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Comments

a(n) = 0 iff n is a perfect power m^k, m >= 1, k >= 2 (i.e., a member of A001597).

Of course if n is a perfect power then a(n) = 0, but it seems that the other direction is true only assuming the generalized Artin's conjecture. See the link from Tomás Oliveira e Silva below. - Jianing Song, Jan 22 2019

Examples

			a(2) = 3, since 3 has 2 as smallest positive primitive root and no prime p < 3 has 2 as smallest positive primitive root.
a(24) = 533821, since prime 533821 has 24 as smallest positive primitive root and no prime p < 533821 has 24 as smallest positive primitive root.
		

References

• 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. XLIV.

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..107 (from the web page of Tomás Oliveira e Silva)
• Wouter Meeussen, Smallest Primes with Specified Least Primitive Root
• Tomás Oliveira e Silva, Least primitive root of prime numbers
• Index entries for primes by primitive root

Crossrefs

Cf. A001122-A001126, A061323-A061335, A061730-A061741.

Indices of the primes: A066529.

For records see A133433. See A133432 for a version without the 0's.

Programs

Mathematica

t = Table[0, {100}]; Do[a = PrimitiveRoot@Prime@n; If[a < 101 && t[[a]] == 0, t[[a]] = n], {n, 10^6}]; Unprotect[Prime]; Prime[0] = 0; Prime@t; Clear[Prime]; Protect[Prime] (* Robert G. Wilson v, Dec 15 2005 *)

Python

from sympy import nextprime, perfect_power, primitive_root
def a(n):
    if perfect_power(n): return 0
    p = 2
    while primitive_root(p) != n: p = nextprime(p)
    return p
print([a(n) for n in range(1, 40)]) # Michael S. Branicky, Feb 13 2023

Python

# faster version for initial segment of sequence
from itertools import count, islice
from sympy import nextprime, perfect_power, primitive_root
def agen(): # generator of terms
    p, adict, n = 2, {None: 0}, 1
    for k in count(1):
        v = primitive_root(p)
        if v not in adict:
            adict[v] = p
        if perfect_power(n): adict[n] = 0
        while n in adict: yield adict[n]; n += 1
        p = nextprime(p)
print(list(islice(agen(), 40))) # Michael S. Branicky, Feb 13 2023

Formula

a(n) = min { prime(k) | A001918(k) = n } U {0} = A000040(A066529(n)) (or zero). - M. F. Hasler, Jun 01 2018

Extensions

Comment corrected by Christopher J. Smyth, Oct 16 2013