A002230 Primes with record values of the least positive primitive root.

2, 3, 7, 23, 41, 71, 191, 409, 2161, 5881, 36721, 55441, 71761, 110881, 760321, 5109721, 17551561, 29418841, 33358081, 45024841, 90441961, 184254841, 324013369, 831143041, 1685283601, 6064561441, 7111268641, 9470788801, 28725635761, 108709927561, 386681163961, 1990614824641, 44384069747161, 89637484042681

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

References

• 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).
• 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

• Michel Marcus, Table of n, a(n) for n = 1..38 (using McGown and Sorenson).
• Stephen D. Cohen, Tomás Oliveira e Silva, and Tim Trudgian, On Grosswald's conjecture on primitive roots, arXiv:1503.04519 [math.NT], 2015.
• R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
• Kevin J. McGown and Jonathan P. Sorenson, Computation of the least primitive root, arXiv:2206.14193 [math.NT], 2022.
• Tomás Oliveira e Silva, Least prime primitive root of prime numbers
• 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]
• Index entries for primes by primitive root

Crossrefs

Cf. A002229 (for the primitive roots in question).

Records in A023048, indices in A114885.

Programs

Mathematica

s = {2}; rm = 1; Do[p = Prime[k]; r = PrimitiveRoot[p]; If[r > rm, Print[p]; AppendTo[s, p]; rm = r], {k, 10^6}]; s (* Jean-François Alcover, Apr 05 2011 *)
DeleteDuplicates[Table[{p,PrimitiveRoot[p,1]},{p,Prime[Range[61100]]}],GreaterEqual[ #1[[2]],#2[[2]]]&][[All,1]] (* The program generates the first 15 terms of the sequence. *) (* Harvey P. Dale, Aug 22 2022 *)

Python

from sympy import isprime, primitive_root
from itertools import count, islice
def f(n): return 0 if not isprime(n) or (r:=primitive_root(n))==None else r
def agen(r=0): yield from ((m, r:=f(m))[0] for m in count(1) if f(m) > r)
print(list(islice(agen(), 15))) # Michael S. Branicky, Feb 13 2023

Extensions

More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu)