A001126 -id:A001126 - cp's OEIS frontend

A061323 Primes with 10 as smallest positive primitive root.

313, 337, 1021, 1297, 1783, 1873, 2137, 2971, 3221, 3313, 4051, 4339, 5233, 5531, 5743, 6073, 6301, 6337, 6553, 6793, 7177, 7753, 8233, 9109, 9697, 9829, 9931, 10273, 10781, 11059, 11149, 11257, 11617, 11941, 11971, 12143, 12457, 12577

Offset: 1

Klaus Brockhaus, Apr 24 2001

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A001918, A001126, A001913.

Prime[ Select[ Range[2000], PrimitiveRoot[ Prime[ # ] ] == 10 & ] ]
Select[ Prime@Range@1510, PrimitiveRoot@# == 10 &] (* Robert G. Wilson v, May 11 2001 *)

is(n)=n>9 && isprime(n) && znorder(Mod(2,n))Charles R Greathouse IV, Apr 24 2015

A061335 Primes with 23 as smallest positive primitive root.

Original entry on oeis.org

2161, 8761, 10559, 12479, 12911, 13729, 15601, 18121, 19009, 21787, 31249, 35281, 37321, 42841, 43201, 49921, 50951, 51239, 52711, 53231, 67489, 70249, 79801, 88919, 90121, 90289, 91393, 97919, 106129, 106391, 106681, 107881, 108529, 115201, 123191, 126311

Offset: 1

Klaus Brockhaus, Apr 25 2001

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A001918, A001126, A019350.

Prime[ Select[ Range[12100], PrimitiveRoot[ Prime[ # ] ] == 23 & ] ]
Select[ Prime@Range@10880, PrimitiveRoot@# == 23 &] (* Robert G. Wilson v, May 11 2001 *)

is(n)=my(v=[2, 3, 5, 7, 11, 13, 17, 19, 6, 10, 12, 14, 15, 18, 20, 21, 22]); if(!isprime(n) || n<99, return(0)); for(i=1,#v, if(znorder(Mod(v[i], n))==n-1, return(0))); znorder(Mod(23,n))==n-1 \\ Charles R Greathouse IV, Apr 27 2015

A061730 Primes with 24 as smallest positive primitive root.

Original entry on oeis.org

533821, 567631, 672181, 843781, 1035301, 1512421, 1929061, 2260501, 2839621, 2894431, 2896741, 4466221, 5428231, 5970511, 6170911, 9340501, 9730711, 9920821, 10635661, 10684759, 10720711, 10870471, 11425261, 11591581

Offset: 1

Klaus Brockhaus, May 06 2001

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..225

Cf. A001918, A001126, A019351.

Prime[ Select[ Range[ 10^6], PrimitiveRoot[ Prime[ # ] ] == 24 & ] ]
(* or *)
Select[ Prime@Range@1000000, PrimitiveRoot@# == 24 &] (* Robert G. Wilson v, May 11 2001 *)

is(n)=if(n<9||!isprime(n), return(0)); for(k=2,23, if(znorder(Mod(k,n))==n-1, return(0))); znorder(Mod(24,n))==n-1 \\ Charles R Greathouse IV, Apr 28 2015

More terms from Robert G. Wilson v, May 11 2001

A061741 Primes with 39 as smallest positive primitive root.

Original entry on oeis.org

166031, 264961, 325249, 388081, 450071, 462841, 543601, 735271, 816649, 823201, 915049, 1063561, 1155151, 1414081, 1415929, 1554169, 1704271, 1884121, 1952449, 2181271, 2215921, 2290831, 2477521, 2499421, 2514961, 2585647, 2633689

Offset: 1

Klaus Brockhaus, May 06 2001

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..660

Cf. A001918, A019364, A001122-A001126, A061323-A061335, A061730-A061741, A114657-A114686.

Select[ Prime@Range@221000, PrimitiveRoot@# == 39 &] (* Robert G. Wilson v, May 11 2001 *)

is(n)=if(n<9||!isprime(n), return(0)); for(k=2,38,if(znorder(Mod(k,n))==n-1, return(0))); znorder(Mod(39,n))==n-1 \\ Charles R Greathouse IV, Apr 28 2015

More terms from Robert G. Wilson v, May 11 2001 and Dec 21 2005

A114657 Primes with 40 as smallest positive primitive root.

Original entry on oeis.org

1373989, 3296581, 3771211, 4014739, 4073371, 5191033, 15188731, 19461661, 23108101, 27478621, 27945061, 39242701, 40393051, 48942661, 51113941, 60043411, 62362411, 66868621, 71443531, 73572181, 94008091, 103767691, 106066171, 110543581, 110950171, 114407101

Offset: 1

Robert G. Wilson v, Dec 21 2005

nonn

Robert Price, Table of n, a(n) for n = 1..302

Cf. A001122-A001126, A061323-A061335, A061730-A061741, A114657-A114686.

Select[ Prime@Range@6354000, PrimitiveRoot@# == 40 &]

a(23)-a(26) from Robert Price, Nov 18 2023

A114686 Primes with 71 as smallest positive primitive root.

Original entry on oeis.org

3659401, 8453041, 10319761, 14155681, 16391761, 18094561, 19616689, 20456329, 21677041, 22628929, 27275161, 32051881, 34228489, 37728601, 38884561, 39191881, 40101071, 40167241, 42163969, 47931601, 48461449, 49460161, 50389441, 54932329, 56219281, 57590569

Offset: 1

Robert G. Wilson v, Dec 21 2005

nonn

Robert Price, Table of n, a(n) for n = 1..1366

Cf. A001122-A001126, A061323-A061335, A061730-A061741, A114657-A114686.

t={}; Do[ If[ PrimitiveRoot[ Prime@n] == 71, AppendTo[t, n]; Print@ Prime@n], {n, 3280000}]; Prime@t

is(n)=if(n<72,return(0));for(k=2,70,if(znorder(Mod(k,n))==n-1,return(0)));znorder(Mod(71,n))==n-1&&isprime(n) \\ Charles R Greathouse IV, Jul 19 2011

is(n)=isprime(n)&&lift(znprimroot(n))==71 \\ relies on implementation details, may not always work
\\ Charles R Greathouse IV, Jul 19 2011

a(23) and beyond from Robert Price, Nov 20 2023

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

A061325 Primes with 12 as smallest positive primitive root.

Original entry on oeis.org

4111, 7841, 10111, 15391, 15991, 16061, 20011, 21031, 22699, 32299, 32957, 35911, 43963, 45127, 45631, 47431, 49831, 51199, 53731, 58111, 59671, 60331, 64231, 74311, 76039, 78079, 81331, 81761, 83311, 83431, 98911, 102871, 104729, 108907

Offset: 1

Klaus Brockhaus, Apr 24 2001

nonn

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A001918, A001126, A019340.

Prime[ Select[ Range[15000], PrimitiveRoot[ Prime[ # ] ] == 12 & ] ]
(* or *)
Select[ Prime@Range@15000, PrimitiveRoot@# == 12 &] (* Robert G. Wilson v, May 11 2001 *)

A061330 Primes with 18 as smallest positive primitive root.

Original entry on oeis.org

53173, 87541, 100621, 124909, 137341, 164341, 192611, 226549, 230101, 241861, 260317, 262909, 288661, 309541, 352309, 371029, 425701, 458701, 461891, 463741, 476029, 490741, 491461, 562021, 569869, 627661, 640069, 661621, 664141, 690541

Offset: 1

Klaus Brockhaus, Apr 25 2001

nonn

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A001918, A001126, A019345.

Prime[ Select[ Range[100000], PrimitiveRoot[ Prime[ # ] ] == 18 & ] ]
(* or *)
Select[ Prime@Range@100000, PrimitiveRoot@# == 18 &] (* Robert G. Wilson v *)

More terms from Robert G. Wilson v, May 11 2001

A114659 Primes with 42 as smallest positive primitive root.

Original entry on oeis.org

2494381, 3835651, 5750071, 6610561, 9014281, 9288121, 9781201, 11350561, 12618061, 14573761, 16576981, 17640871, 22293241, 23513311, 24793921, 24885271, 27665821, 29904601, 30824401, 30895231, 31697821, 33515701, 38340901, 39033721, 39037351, 39051841

Offset: 1

Robert G. Wilson v, Dec 21 2005

nonn

Robert Price, Table of n, a(n) for n = 1..1040

Cf. A001122-A001126, A061323-A061335, A061730-A061741, A114657-A114686.

(* first load *) << NumberTheory`NumberTheoryFunctions` (* then *) Select[ Prime@Range@2379000, PrimitiveRoot@# == 42 &]

a(23)-a(26) from Robert Price, Nov 19 2023

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A061323 Primes with 10 as smallest positive primitive root.

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A061335 Primes with 23 as smallest positive primitive root.

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A061730 Primes with 24 as smallest positive primitive root.

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A061741 Primes with 39 as smallest positive primitive root.

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A114657 Primes with 40 as smallest positive primitive root.

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A114686 Primes with 71 as smallest positive primitive root.

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

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A061325 Primes with 12 as smallest positive primitive root.

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