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A023048
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Smallest prime having least positive primitive root n, or 0 if no such prime exists.
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11
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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
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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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
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FORMULA
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a(n) = min { prime(k) | A001918(k) = n } U {0} = A000040(A066529(n)) (or zero). - M. F. Hasler, Jun 01 2018
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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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.
Sequence in context: A203143 A249523 A301316 * A083521 A104691 A011160
Adjacent sequences: A023045 A023046 A023047 * A023049 A023050 A023051
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KEYWORD
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nonn
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AUTHOR
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David W. Wilson
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EXTENSIONS
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Comment corrected by Christopher J. Smyth, Oct 16 2013
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STATUS
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approved
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