A133433 -id:A133433 - cp's OEIS frontend

A318239 a(n) is the smallest primitive root of A133433(n).

2, 3, 5, 6, 7, 10, 11, 12, 18, 20, 24, 40, 42, 45, 48, 54, 72, 80, 96

Offset: 1

Jianing Song, Aug 22 2018

Also with an initial 1, numbers k such that A023048(k) set a new record.
If A023048(108) != 0 (which is implied assuming generalized Artin's conjecture) then the next term of this sequence is 108.
Conjecturally 11 is the largest prime in this sequence, 42 is the largest squarefree term, and 45 is the largest odd term.

			The smallest prime with least primitive root 11 is 643, and the smallest prime with least primitive root 2, 3, 5, ..., 10 are all < 643, so 11 is a term.
The smallest prime with least primitive root 45 is 95525767, and the smallest prime with least primitive root 2, 3, 5, ..., 44 are all < 95525767, so 45 is a term.
13 is not a term since the smallest prime with least primitive root 13 is 457, but the smallest prime with least primitive root 12 is 4111, which is larger than 457.

Tomás Oliveira e Silva, Least primitive root of prime numbers

Cf. A023048, A133433.

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

Original entry on oeis.org

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

David W. Wilson

nonn

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

			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.

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.

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

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.

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 *)

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

# 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

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

Comment corrected by Christopher J. Smyth, Oct 16 2013

A133432 Let m = n-th number that is not a perfect power, A007916(n). Then a(n) = smallest prime having least positive primitive root m.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Nov 29 2007

nonn

a(n) = A023048(A007916(n)).

N. J. A. Sloane, Table of n, a(n) for n=1..94 (from the web page of Tomás Oliveira e Silva)
Tomás Oliveira e Silva, Least primitive root of prime numbers

Cf. A023048, A007916, A001597, A133433 (records).

Definition corrected by Christopher J. Smyth, Oct 16 2013

cp's OEIS Frontend

A318239 a(n) is the smallest primitive root of A133433(n).

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

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A133432 Let m = n-th number that is not a perfect power, A007916(n). Then a(n) = smallest prime having least positive primitive root m.

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