A001122 -id:A001122 - cp's OEIS frontend

A001917 (p-1)/x, where p = prime(n) and x = ord(2,p), the smallest positive integer such that 2^x == 1 (mod p).

1, 1, 2, 1, 1, 2, 1, 2, 1, 6, 1, 2, 3, 2, 1, 1, 1, 1, 2, 8, 2, 1, 8, 2, 1, 2, 1, 3, 4, 18, 1, 2, 1, 1, 10, 3, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 6, 1, 3, 8, 2, 10, 5, 16, 2, 1, 2, 3, 4, 3, 1, 3, 2, 2, 1, 11, 16, 1, 1, 4, 2, 2, 1, 1, 2, 1, 9, 2, 2, 1, 1, 10, 6, 6, 1, 2, 6, 1, 2, 1, 2, 2, 1, 3, 2, 1, 2, 1, 1, 1, 1, 1, 2

Offset: 2

N. J. A. Sloane

Also number of cycles in permutations constructed from siteswap juggling pattern 1234...p.
Also the number of irreducible polynomial factors for the polynomial (x^p-1)/(x-1) over GF(2), where p is the n-th prime. - V. Raman, Oct 04 2012
The sequence is unbounded: for any value of M, there exists an element of the sequence divisible by M. See the proof by David Speyer below. - Shreevatsa R, May 24 2013

M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 131.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 7-10.
W. Meissner, Über die Teilbarkeit von 2^p-2 durch das Quadrat der Primzahl p = 1093, Sitzungsberichte Königlich Preussischen Akadamie Wissenschaften Berlin, 35 (1913), 663-667.
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).

T. D. Noe, Table of n, a(n) for n = 2..10000
I. Anderson and D. A. Preece, Combinatorially fruitful properties of 3*2^(-1) and 3*2^(-2) modulo p, Discr. Math., 310 (2010), 312-324.
Keith Conrad, Wieferich primes, Univ. Conn. (2023).
W. Meissner, Über die Teilbarkeit von 2^p-2 durch das Quadrat der Primzahl p = 1093, Sitzungsberichte Königlich Preussischen Akadamie Wissenschaften Berlin, 35 (1913), 663-667. [Annotated scanned copy]
V. Papadimitriou, The l_2^(p) and the ... ratio of the first hundred million primes
David Speyer, Can the order of 2 mod p be arbitrarily small (relative to p-1)?

Cf. A006694 gives cycle counts of such permutations constructed for all odd numbers.
Cf. A002323, A001122, A115591, A001133, A001134, A001135, A001136, A101208.
Cf. A014664.

[ (p-1)/Modorder(2, p) where p is NthPrime(n): n in [2..100] ]; // Klaus Brockhaus, Dec 09 2008

with(numtheory); [seq((ithprime(n)-1)/order(2,ithprime(n)),n=2..130)];
with(group); with(numtheory); gen_rss_perm := proc(n) local a, i; a := []; for i from 1 to n do a := [op(a), ((2*i) mod (n+1))]; od; RETURN(a); end; count_of_disjcyc_seq := [seq(nops(convert(gen_rss_perm(ithprime(j)-1),'disjcyc')),j=2..)];

a6694[n_] := Sum[ EulerPhi[d] / MultiplicativeOrder[2, d], {d, Divisors[2n + 1]}] - 1; a[n_] := a6694[(Prime[n]-1)/2]; Table[ a[n], {n, 2, 104}] (* Jean-François Alcover, Dec 14 2011, after Vladimir Shevelev *)
Table[p = Prime[n]; (p - 1)/MultiplicativeOrder[2, p], {n, 2, 100}] (* T. D. Noe, Apr 11 2012 *)
ord[n_]:=Module[{x=1},While[PowerMod[2,x,n]!=1,x++];(n-1)/x]; ord/@ Prime[ Range[ 2,110]] (* Harvey P. Dale, Jun 25 2014 *)

{for(n=2, 100, p=prime(n); print1((p-1)/znorder(Mod(2, p)), ","))} \\ Klaus Brockhaus, Dec 09 2008

from sympy import prime, n_order
def A001917(n):
    p = prime(n)
    return 1 if n == 2 else (p-1)//n_order(2,p) # Chai Wah Wu, Jan 15 2020

From Vladimir Shevelev, May 26 2008: (Start)
a(n) = A006694((p_n-1)/2) where p_n is the n-th odd prime.
Conjecture: k*a(n) = A006694(((p_n)^k-1)/2). (End)

Additional comments from Antti Karttunen, Jan 05 2000

More terms from N. J. A. Sloane, Dec 24 2009

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

A167791 Numbers with primitive root 2.

Original entry on oeis.org

3, 5, 9, 11, 13, 19, 25, 27, 29, 37, 53, 59, 61, 67, 81, 83, 101, 107, 121, 125, 131, 139, 149, 163, 169, 173, 179, 181, 197, 211, 227, 243, 269, 293, 317, 347, 349, 361, 373, 379, 389, 419, 421, 443, 461, 467, 491, 509, 523, 541, 547, 557, 563, 587, 613, 619

Offset: 1

T. D. Noe, Nov 12 2009

nonn

Numbers k such that the binary expansion of 1/k has period phi(k). For example 1/27 has a period of 18 bits.
All entries are odd. An odd composite number n can have a primitive root if and only if it is a prime power (see A033948). - V. Raman, Oct 04 2012
It is unknown whether there is a prime p such that p is in this sequence while p^2 is not. - Jianing Song, Jan 27 2019

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)

Cf. A000010, A001122 (primes with primitive root 2), A033948.

[n: n in [3..619] | IsPrimitive(2, n)]; // Arkadiusz Wesolowski, Dec 22 2020

pr=2; Select[Range[2,2000], MultiplicativeOrder[pr,# ] == EulerPhi[ # ] &]

for(n=3,200,if(n%2==1&&znorder(Mod(2,n))==eulerphi(n),printf(n","))) \\ V. Raman, Oct 04 2012

is(n)=n%2 && isprimepower(n) && znorder(Mod(2,n))==eulerphi(n-1) \\ Charles R Greathouse IV, Jul 05 2013

A019334 Primes with primitive root 3.

Original entry on oeis.org

2, 5, 7, 17, 19, 29, 31, 43, 53, 79, 89, 101, 113, 127, 137, 139, 149, 163, 173, 197, 199, 211, 223, 233, 257, 269, 281, 283, 293, 317, 331, 353, 379, 389, 401, 449, 461, 463, 487, 509, 521, 557, 569, 571, 593, 607, 617, 631, 641, 653, 677, 691, 701, 739, 751, 773, 797

Offset: 1

David W. Wilson

nonn

To allow primes less than the specified primitive root m (here, 3) to be included, we use the essentially equivalent definition "Primes p such that the multiplicative order of m mod p is p-1". This comment applies to all of A019334-A019421. - N. J. A. Sloane, Dec 02 2019
From Jianing Song, Apr 27 2019: (Start)
All terms except the first are congruent to 5 or 7 modulo 12. If we define
  Pi(N,b) = # {p prime, p <= N, p == b (mod 12)};
     Q(N) = # {p prime, 2 < p <= N, p in this sequence},
then by Artin's conjecture, Q(N) ~ C*N/log(N) ~ 2*C*(Pi(N,5) + Pi(N,7)), where C = A005596 is Artin's constant.
If we further define
   Q(N,b) = # {p prime, p <= N, p == b (mod 12), p in this sequence},
then we have:
   Q(N,5) ~ (3/5)*Q(N) ~ (12/5)*C*Pi(N,5);
   Q(N,7) ~ (2/5)*Q(N) ~ ( 8/5)*C*Pi(N,7).
For example, for the first 1000 terms except for a(1) = 2, there are 593 terms == 5 (mod 12) and 406 terms == 7 (mod 12). (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
J. Conde, M. Miller, J. M. Miret, K. Saurav, On the Nonexistence of Almost Moore Digraphs of Degree Four and Five, International Conference on Mathematical Computer Engineering (ICMCE-13), pp. 2-7, At VIT University, Chennai, Volume: I, 2013.
J. Conde, M. Miller, J. M. Miret, K. Saurav, On the Nonexistence of Almost Moore Digraphs of Degree Four and Five, Mathematics in Computer Science, June 2015, Volume 9, Issue 2, pp 145-149.
Eric Weisstein's World of Mathematics, Artin's constant
Wikipedia, Artin's conjecture on primitive roots
Index entries for primes by primitive root

Cf. A005596, A001122 (primitive root 2).

Cf. A019335-A019421.

pr=3; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &]

isok(p) = isprime(p) && (p!=3) && (znorder(Mod(3, p))+1 == p); \\ Michel Marcus, May 12 2019

A006883 Long period primes: the decimal expansion of 1/p has period p-1.

Original entry on oeis.org

2, 7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, 223, 229, 233, 257, 263, 269, 313, 337, 367, 379, 383, 389, 419, 433, 461, 487, 491, 499, 503, 509, 541, 571, 577, 593, 619, 647, 659, 701, 709, 727, 743, 811, 821, 823, 857, 863

Offset: 1

Robert Munafo

Also called full reptend primes or maximal period primes.
Also called golden primes or long primes.
Here, as opposed to A001913, 2 is a term, because the decimal expansion of 1/2 is 0.5000000000..., so it is periodic with period 1 and pattern 0. - Michel Marcus, Jun 06 2018

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 864.
Albert H. Beiler, Recreations in the Theory of Numbers, 2nd ed. New York: Dover, 1966, pages 65, 309.
John H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, p. 161.
Carl Friedrich Gauss, "Disquisitiones Arithmeticae"
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 115.
M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 61.
D. H. Lehmer, A note on primitive roots, Scripta Mathematica, vol. 26 (1963), p. 117. [Gives some interesting information about the frequency of maximal period primes and discusses two freak cases.]
C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory, Oxford University Press, 1966, pp. 56-58.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Eric Weisstein's World of Mathematics, Full Reptend Prime.
Index entries for sequences related to decimal expansion of 1/n

Apart from initial term, identical to A001913.
Cf. A005596, A006559, A067556.
Cf. A001122 (long period primes in binary).

isA006883 := proc(p) if p = 2 then true; elif isprime(p) then RETURN( numtheory[order](10,p) = p-1) ; else false; fi; end: for i from 1 to 300 do p := ithprime(i) ; if isA006883(p) then printf("%d ",p) ; fi; od: # R. J. Mathar, Apr 01 2009

f[n_Integer] := Block[{ds = Divisors[n - 1]}, (n - 1)/Take[ ds, Position[ PowerMod[ 10, ds, n], 1] [[1, 1]]] [[ -1]]]; Select[ Prime[ Range[4, 150]], f[ # ] == 1 &] (* Robert G. Wilson v, Sep 14 2004 *)
maxPeriodQ[p_] := MultiplicativeOrder[10, p] == p-1; maxPeriodQ[2] = True; Select[ Prime[ Range[150]], maxPeriodQ] (* Jean-François Alcover, Jan 07 2013 *)

print1(2);forprime(p=7,1e3,if(znorder(Mod(10,p))+1==p,print1(", "p))) \\ Charles R Greathouse IV, Feb 27 2011

From Gerard Schildberger, Jul 02 2005: (Start)
Emil Artin conjectured that the proportion of primes that belong to this sequence can be expressed as:
(2*1-1)(3*2-1)(5*4-1)(7*6-1)(11*10-1)(13*12-1)...
------------------------------------------------- = 0.373955813619202288...
(2*1)(3*2)(5*4)(7*6)(11*10)(13*12)...
(End)
This Artin's constant, Product_{p prime} (1-1/(p^2-p)), is referenced in A005596. - Robert FERREOL, Jun 05 2018

More terms from James Sellers, Aug 21 2000

Additional comments from Jason Earls, Apr 06 2001

A112927 a(n) is the least prime such that the multiplicative order of 2 mod a(n) equals n, or a(n)=1 if no such prime exists.

Original entry on oeis.org

1, 3, 7, 5, 31, 1, 127, 17, 73, 11, 23, 13, 8191, 43, 151, 257, 131071, 19, 524287, 41, 337, 683, 47, 241, 601, 2731, 262657, 29, 233, 331, 2147483647, 65537, 599479, 43691, 71, 37, 223, 174763, 79, 61681, 13367, 5419, 431, 397, 631, 2796203, 2351, 97, 4432676798593, 251, 103, 53, 6361, 87211

Offset: 1

Vladimir Shevelev, Aug 25 2008

nonn

If a(n) differs from 1, then a(n) is the minimal prime divisor of A064078(n);
a(n)=n+1 iff n+1 is prime from A001122; a(n)=2n+1 iff 2n+1 is prime from A115591.
If a(n) > 1 then a(n) is the index where n occurs first in A014664. - M. F. Hasler, Feb 21 2016
Bang's theorem (special case of Zsigmondy's theorem, see links): a(n)>1 for all n>6. - Jeppe Stig Nielsen, Aug 31 2020

Robert G. Wilson v, Table of n, a(n) for n = 1..1206
Dario Alejandro Alpern, Factorization using the Elliptic Curve Method
Will Edgington, Factored Mersenne Numbers [from Internet Archive Wayback Machine]
Wikipedia, Zsigmondy's theorem

Cf. A002326, A064078, A001122, A115591.

Cf. A112927 (base 2), A143663 (base 3), A112092 (base 4), A143665 (base 5), A379639 (base 6), A379640 (base 7), A379641 (base 8), A379642 (base 9), A007138 (base 10), A379644 (base 11), A252170 (base 12).

A112927(n,f=factor(2^n-1)[,1])=!for(i=1,#f,znorder(Mod(2,f[i]))==n&&return(f[i])) \\ Use the optional 2nd arg to give a list of pseudoprimes to try when factoring of 2^n-1 is too slow. You may try factor(2^n-1,0)[,1]. - M. F. Hasler, Feb 21 2016

A023212 Primes p such that 4*p+1 is also prime.

Original entry on oeis.org

3, 7, 13, 37, 43, 67, 73, 79, 97, 127, 139, 163, 193, 199, 277, 307, 373, 409, 433, 487, 499, 577, 619, 673, 709, 727, 739, 853, 883, 919, 997, 1033, 1039, 1063, 1087, 1093, 1123, 1129, 1297, 1327, 1423, 1429, 1453, 1543, 1549, 1567, 1579, 1597, 1663, 1753

Offset: 1

David W. Wilson

nonn

If p > 3 is a Sophie Germain prime (A005384), p cannot be in this sequence, because all Germain primes greater than 3 are of the form 6k - 1, and then 4p + 1 = 3*(8k-1). - Enrique Pérez Herrero, Aug 15 2011
a(n), except 3, is of the form 6k+1. - Enrique Pérez Herrero, Aug 16 2011
According to Beiler: the integer 2 is a primitive root of all primes of the form 4p + 1 with p prime. - Martin Renner, Nov 06 2011
Chebyshev showed that 2 is a primitive root of all primes of the form 4p + 1 with p prime. - Jonathan Sondow, Feb 04 2013
Also solutions to the equation: floor(4/A000005(4*n^2+n)) = 1. - Enrique Pérez Herrero, Jan 12 2013
Prime numbers p such that p^p - 1 is divisible by 4*p + 1. - Gary Detlefs, May 22 2013
It appears that whenever (p^p - 1)/(4*p + 1) is an integer, then this integer is even (see previous comment). - Alexander R. Povolotsky, May 23 2013
4p + 1 does not divide p^n + 1 for any n. - Robin Garcia, Jun 20 2013
Primes in this sequence of the form 4k+1 are listed in A113601. - Gary Detlefs, May 07 2019
There are no numbers with last digit 1 in this list (i.e., members of A030430) because primes p == 1 (mod 10) lead to 5|(4p+1) such that 4p+1 is not prime. - R. J. Mathar, Aug 13 2019

Albert H. Beiler, Recreations in the theory of numbers, New York: Dover, (2nd ed.) 1966, p. 102, nr. 5.
P. L. Chebyshev, Theory of congruences, Elements of number theory, Chelsea, 1972, p. 306.

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
Grigory Ryabov, On schurity of the dihedral group D_(2p), arXiv:2308.14209 [math.GR], 2023.
Rosemary Sullivan and Neil Watling, Independent divisibility pairs on the set of integers from 1 to n, INTEGERS, Vol. 13 (2013), Article A65.
Samuel S. Wagstaff, Jr., Sum of Reciprocals of Germain Primes, Journal of Integer Sequences, Vol. 24, No. 2 (2021), Article 21.9.5.

Cf. A001122, A005384, A043297, A088730.
Cf. A005098, A090866.
Cf. A182265, A182434.

[n: n in [0..1000] | IsPrime(n) and IsPrime(4*n+1)]; // Vincenzo Librandi, Nov 20 2010

isA023212 := proc(n)
    isprime(n) and isprime(4*n+1) ;
end proc:
for n from 1 to 1800 do
    if isA023212(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, May 26 2013

Select[Range[2000], PrimeQ[#] && PrimeQ[4# + 1] &] (* Alonso del Arte, Aug 15 2011 *)
Join[{3}, Select[Range[7, 2000, 6], PrimeQ[#] && PrimeQ[4# + 1] &]] (* Zak Seidov, Jan 21 2012 *)
Select[Prime[Range[300]],PrimeQ[4#+1]&] (* Harvey P. Dale, Oct 17 2021 *)

forprime(p=2,1800,if(Mod(p,4*p+1)^p==1, print1(p", \n"))) \\ Alexander R. Povolotsky, May 23 2013

Sum_{n>=1} 1/a(n) is in the interval (0.892962433, 1.1616905) (Wagstaff, 2021). - Amiram Eldar, Nov 04 2021

Name edited by Michel Marcus, Nov 27 2020

A115591 Primes p such that the multiplicative order of 2 modulo p is (p-1)/2.

Original entry on oeis.org

7, 17, 23, 41, 47, 71, 79, 97, 103, 137, 167, 191, 193, 199, 239, 263, 271, 311, 313, 359, 367, 383, 401, 409, 449, 463, 479, 487, 503, 521, 569, 599, 607, 647, 719, 743, 751, 761, 769, 809, 823, 839, 857, 863, 887, 929, 967, 977, 983, 991, 1009, 1031

Offset: 1

Don Reble, Mar 11 2006

nonn

It appears that this is also the sequence of values of n for which the sum of terms of one period of the base-2 MR-expansion (see A136042) of 1/n equals (n-1)/2. An example appears in A155072 where one period of the base-2 MR-expansion of 1/17 is shown to be {5,1,1,1} with sum 8=(17-1)/2. - John W. Layman, Jan 19 2009

If p is a term of this sequence, then 2 is a quadratic residue module p, so p == 1, 7 (mod 8). - Jianing Song, Nov 01 2024

Klaus Brockhaus, Table of n, a(n) for n=1..1000

Cf. A001122, A001133.

Cf. A136042, A155072. - John W. Layman, Jan 19 2009

[ p: p in PrimesUpTo(1031) | r eq 1 and Order(R!2) eq q where q,r is Quotrem(p,2) where R is ResidueClassRing(p) ]; // Klaus Brockhaus, Dec 02 2008

fQ[n_] := 1 + 2 MultiplicativeOrder[2, n] == n; Select[ Prime@ Range@ 174, fQ]

r=2;forprime(p=3,1500,z=(p-1)/znorder(Mod(r,p));if(z==2,print1(p,", "))); \\ Joerg Arndt, Jan 12 2011

cp's OEIS Frontend

A001917 (p-1)/x, where p = prime(n) and x = ord(2,p), the smallest positive integer such that 2^x == 1 (mod p).

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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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A167791 Numbers with primitive root 2.

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A019334 Primes with primitive root 3.

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A006883 Long period primes: the decimal expansion of 1/p has period p-1.

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