A060749 -id:A060749 - cp's OEIS frontend

A001918 Least positive primitive root of n-th prime.

1, 2, 2, 3, 2, 2, 3, 2, 5, 2, 3, 2, 6, 3, 5, 2, 2, 2, 2, 7, 5, 3, 2, 3, 5, 2, 5, 2, 6, 3, 3, 2, 3, 2, 2, 6, 5, 2, 5, 2, 2, 2, 19, 5, 2, 3, 2, 3, 2, 6, 3, 7, 7, 6, 3, 5, 2, 6, 5, 3, 3, 2, 5, 17, 10, 2, 3, 10, 2, 2, 3, 7, 6, 2, 2, 5, 2, 5, 3, 21, 2, 2, 7, 5, 15, 2, 3, 13, 2, 3, 2, 13, 3, 2, 7, 5, 2, 3, 2, 2, 2, 2, 2, 3

Offset: 1

N. J. A. Sloane

If k is a primitive root of p=4m+1, then p-k is too. If k is a primitive root of p=4m+3, then p-k isn't, but has order 2m+1. - Jon Perry, Sep 07 2014

			modulo 7: 3^6=1, 3^2=2, 3^7=3, 3^4=4, 3^5=5, 3^3=6, 7=prime(4), 3=a(4).

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.
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 213.
CRC Handbook of Combinatorial Designs, 1996, p. 615.
P. Fan and M. Darnell, Sequence Design for Communications Applications, Wiley, NY, 1996, Table A.1.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 111.
Hua Loo Keng, Introduction To Number Theory, 'Table of least primitive roots for primes less than 50000', pp. 52-6, Springer NY 1982.
R. Osborn, Tables of All Primitive Roots of Odd Primes Less Than 1000, Univ. Texas Press, 1961.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 20.
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).

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
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].
Loo-keng Hua, On the least primitive root of a prime, Bull. Amer. Math. Soc. 48 (1942), 726-730.
K. Matthews, Finding the least primitive root (mod p), p an odd prime
Jorma K. Merikoski, Pentti Haukkanen, and Timo Tossavainen, The congruence x^n = -a^n (mod m): Solvability and related OEIS sequences, Notes. Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 516-529. See p. 519.
T. Oliveira e Silva, Least primitive root of prime numbers.
Eric Weisstein's World of Mathematics, Primitive Root.

A column of A060749. Cf. A002233.

A001918 := proc(n)
        numtheory[primroot](ithprime(n)) ;
end proc:

Table[PrimitiveRoot@Prime@n, {n, 101}] (* Robert G. Wilson v, Dec 15 2005 *)
PrimitiveRoot[Prime[Range[110]]] (* Harvey P. Dale, Jan 13 2013 *)

for(x=1, 1000, print1(lift(znprimroot(prime(x))), ", "))

from sympy import prime
from sympy.ntheory.residue_ntheory import primitive_root
def A001918(n): return primitive_root(prime(n)) # Chai Wah Wu, Sep 13 2022

[primitive_root(p) for p in primes(570)] # Zerinvary Lajos, May 24 2009

A088144 Sum of primitive roots of n-th prime.

Original entry on oeis.org

1, 2, 5, 8, 23, 26, 68, 57, 139, 174, 123, 222, 328, 257, 612, 636, 886, 488, 669, 1064, 876, 1105, 1744, 1780, 1552, 2020, 1853, 2890, 1962, 2712, 2413, 3536, 4384, 3335, 5364, 3322, 3768, 4564, 7683, 7266, 8235, 4344, 8021, 6176, 8274

Offset: 1

Ed Pegg Jr, Nov 03 2003

nonn

It is a result that goes back to Mirsky that the set of primes p for which p-1 is squarefree has density A, where A denotes the Artin constant (A = Product_{q prime} (1-1/(q*(q-1)))). Numerically A = 0.3739558136.. = A005596. More precisely, Sum_{p <= x} mu(p-1)^2 = Ax/log x + o(x/log x) as x tends to infinity. Conjecture: sum_{p <= x, mu(p-1)=1} 1 = (A/2)x/log x + o(x/log x) and sum_{p <= x, mu(p-1)=-1} 1 = (A/2)x/log x + o(x/log x). - Pieter Moree (moree(AT)mpim-bonn.mpg.de), Nov 03 2003
The number of the primitive roots is A008330(n). - R. K. Guy, Feb 25 2011
If prime(n)  == 1 (mod 4), then a(n) = prime(n)*A008330(n)/2. There are also primes of the form prime(n) == 3 (mod 4) where prime(n) | a(n), namely prime(n) = 19, 127, 151, 163, 199, 251,... The list of primes in both modulo-4 classes where prime(n)|a(n) is 5, 13, 17, 19, 29, 37, 41, 53, 61,... - R. K. Guy, Feb 25 2011
a(n) = A076410(n) at n = 1, 3, 7, 55,... (for p = 2, 5, 17, 257... and perhaps only for the Fermat primes). - R. K. Guy, Feb 25 2011

			For 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, the primitive roots are as follows: {{1}, {2}, {2, 3}, {3, 5}, {2, 6, 7, 8}, {2, 6, 7, 11}, {3, 5, 6, 7, 10, 11, 12, 14}, {2, 3, 10, 13, 14, 15}, {5, 7, 10, 11, 14, 15, 17, 19, 20, 21}, {2, 3, 8, 10, 11, 14, 15, 18, 19, 21, 26, 27}}

C. F. Gauss, Disquisitiones Arithmeticae, Yale, 1965; see p. 52.

T. D. Noe, Table of n, a(n) for n=1..1000
Leon Mirsky, The Number of Representations of an Integer as the Sum of a Prime and a k-Free Integer, Amer. Math. Monthly 56 (1949), 17-19.

Row sums of A060749, A254309.

Cf. A088145, A121380, A123475, A222009.

PrimitiveRootQ[ a_Integer, p_Integer ] := Block[ {fac, res}, fac = FactorInteger[ p - 1 ]; res = Table[ PowerMod[ a, (p - 1)/fac[ [ i, 1 ] ], p ], {i, Length[ fac ]} ]; ! MemberQ[ res, 1 ] ] PrimitiveRoots[ p_Integer ] := Select[ Range[ p - 1 ], PrimitiveRootQ[ #, p ] & ]
Total /@ Table[PrimitiveRootList[Prime[k]], {k, 1, 45}] (* Updated for Mathematica 13 by Harlan J. Brothers, Feb 27 2023 *)

a(n)=local(r, p, pr, j); p=prime(n); r=vector(eulerphi(p-1)); pr=znprimroot(p); for(i=1, p-1, if(gcd(i, p-1)==1, r[j++]=lift(pr^i))); vecsum(r) \\ after Franklin T. Adams-Watters's code in A060749, Michel Marcus, Mar 16 2015

A216838 Odd primes for which 2 is not a primitive root.

Original entry on oeis.org

7, 17, 23, 31, 41, 43, 47, 71, 73, 79, 89, 97, 103, 109, 113, 127, 137, 151, 157, 167, 191, 193, 199, 223, 229, 233, 239, 241, 251, 257, 263, 271, 277, 281, 283, 307, 311, 313, 331, 337, 353, 359, 367, 383, 397, 401, 409, 431, 433, 439, 449, 457, 463, 479

Offset: 1

V. Raman, Sep 17 2012

nonn

Alternately, for these primes p, the polynomial (x^p+1)/(x+1) is reducible over GF(2).
The prime p belongs to this sequence if and only if A002326((p-1)/2) != (p-1). If A002326((p-1)/2) = (p-1), then the prime p belongs to the sequence A001122. - V. Raman, Dec 01 2012
The only primitive root modulo 2 is 1. See A060749. Hence 2 should be added to this sequence in order to obtain the complement of A001122. - Wolfdieter Lang, May 19 2014

Robert Israel, Table of n, a(n) for n = 1..10000
Anand Bhardwaj, Luisa Degen, Radostin Petkov, and Sidney Stanbury, A Study of Cunningham Bounds through Rogue Primes, arXiv:2311.13375 [math.NT], 2023.

Cf. A002326 (multiplicative order of 2 mod 2n+1)
Cf. A001122 (Primes for which 2 is a primitive root)
Cf. A115586 (Primes for which 2 is neither a primitive root nor a quadratic residue).

select(t -> isprime(t) and numtheory[order](2,t) <> t-1, [seq](2*i+1,i=1..1000)); # Robert Israel, May 20 2014

Select[Prime[Range[2, 100]], PrimitiveRoot[#] =!= 2 &] (* T. D. Noe, Sep 19 2012 *)

forprime(p=3, 1000, if(znorder(Mod(2,p))!=p-1, print(p)))

forprime(p=3, 1000, if(factormod((x^p+1)/(x+1), 2, 1)[1, 1]!=(p-1), print(p)))

Name corrected by Wolfdieter Lang, May 19 2014

A143548 Irregular triangle of numbers k < p^2 such that p^2 divides k^(p-1)-1, with p=prime(n).

Original entry on oeis.org

1, 1, 8, 1, 7, 18, 24, 1, 18, 19, 30, 31, 48, 1, 3, 9, 27, 40, 81, 94, 112, 118, 120, 1, 19, 22, 23, 70, 80, 89, 99, 146, 147, 150, 168, 1, 38, 40, 65, 75, 110, 131, 134, 155, 158, 179, 214, 224, 249, 251, 288, 1, 28, 54, 62, 68, 69, 99, 116, 127, 234, 245, 262, 292, 293, 299, 307, 333, 360

Offset: 1

T. D. Noe, Aug 24 2008

Row n begins with 1 and has prime(n)-1 terms. The first differences of each row are symmetric. For k > p^2, the solutions are just shifted by m*p^2 for m > 0. An open question is whether every integer appears in this sequence. For instance, 2 does not appear until the prime 1093 and 5 does not appear until the prime 20771.
For row n > 1, the sum of the terms in row n is (p-1)*p^2*(p+1)/2, which is A138416. - T. D. Noe, Aug 24 2008, corrected by Robert Israel, Sep 27 2016
Max Alekseyev points out that there is a much faster method of computing these numbers. Let p=prime(n) and let r be a primitive root of p (see A001918 and A060749). Then the terms in row n are r^(k*p) (mod p^2) for k=0..p-2. - T. D. Noe, Aug 26 2008
The numbers in this sequence are the bases to Euler pseudoprimes q, which are squares of prime numbers, such that n^((q-1)/2) == +-1 mod q. An exception is the first number 9 = 3*3, which is, following the strict definition in Crandall and Pomerance, no Fermat pseudoprime and hence no Euler pseudoprime. - Karsten Meyer, Jan 08 2011
For row n > 1, the sum is zero modulo p^2 (rows are antisymmetric due to Binomial Theorem). - Peter A. Lawrence, Sep 11 2016

			(2)   1,
(3)   1, 8,
(5)   1, 7, 18, 24,
(7)   1, 18, 19, 30, 31, 48,
(11)  1, 3, 9, 27, 40, 81, 94, 112, 118, 120,
(13)  1, 19, 22, 23, 70, 80, 89, 99, 146, 147, 150, 168,
(17)  1, 38, 40, 65, 75, 110, 131, 134, 155, 158, 179, 214, 224, 249, 251, 288,

R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2005

T. D. Noe, Rows n=1..50 of triangle, flattened

Cf. A039678, A056020, A056021, A056022, A056024, A056025, A056027, A056028, A056031, A056034, A056035, A096082, A138416.

f:= proc(n) local p,j,x;
  p:= ithprime(n);
  x:= numtheory:-primroot(p);
  op(sort([seq(x^(i*p) mod p^2, i=0..p-2)]))
end proc:
map(f, [$1..20]); # Robert Israel, Sep 27 2016

Flatten[Table[p=Prime[n]; Select[Range[p^2], PowerMod[ #,p-1,p^2]==1&], {n,50}]] (* T. D. Noe, Aug 24 2008 *)
Flatten[Table[p=Prime[n]; r=PrimitiveRoot[p]; b=PowerMod[r,p,p^2]; Sort[NestList[Mod[b*#,p^2]&,1,p-2]], {n,50}]] (* Faster version from T. D. Noe, Aug 26 2008 *)

A071894 Largest positive primitive root (

Original entry on oeis.org

1, 2, 3, 5, 8, 11, 14, 15, 21, 27, 24, 35, 35, 34, 45, 51, 56, 59, 63, 69, 68, 77, 80, 86, 92, 99, 101, 104, 103, 110, 118, 128, 134, 135, 147, 146, 152, 159, 165, 171, 176, 179, 189, 188, 195, 197, 207, 214, 224, 223, 230, 237, 234, 248, 254, 261, 267, 269, 272

Offset: 1

N. J. A. Sloane, Jun 11 2002

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.
R. Osborn, Tables of All Primitive Roots of Odd Primes Less Than 1000, Univ. Texas Press, 1961.

T. D. Noe, Table of n, a(n) for n=1..10000
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].

A diagonal of triangle in A060749.

Cf. A000040, A001918 (least positive primitive root), A002199.

f[n_] := Block[{k = Prime[n] - 1, p = Prime[n], t = Table[i, {i, 1, Prime[n] - 1}]}, While[ Union[ PowerMod[ k, t, p]] != t, k-- ]; k]; Table[ f[n], {n, 1, 60}]

a(n) = my(p=prime(n)); forstep(q=p-1, 1, -1, if(znorder(Mod(q, p))==eulerphi(p), return(q))); \\ Michel Marcus, Sep 28 2023

a(n) = prime(n) - A002199(n) - T. D. Noe, Oct 24 2005

More terms from Robert G. Wilson v, Jun 11 2002

A046147 Triangle read by rows in which row n lists the primitive roots mod n (omitting numbers n without a primitive root).

Original entry on oeis.org

1, 2, 3, 2, 3, 5, 3, 5, 2, 5, 3, 7, 2, 6, 7, 8, 2, 6, 7, 11, 3, 5, 3, 5, 6, 7, 10, 11, 12, 14, 5, 11, 2, 3, 10, 13, 14, 15, 7, 13, 17, 19, 5, 7, 10, 11, 14, 15, 17, 19, 20, 21, 2, 3, 8, 12, 13, 17, 22, 23, 7, 11, 15, 19, 2, 5, 11, 14, 20, 23, 2, 3, 8, 10, 11, 14, 15, 18, 19, 21, 26

Offset: 2

Eric W. Weisstein

			n followed by primitive roots, if any:
1 -
2 1
3 2
4 3
5 2 3
6 5
7 3 5
8 -
9 2 5
10 3 7
11 2 6 7 8
12 -
13 2 6 7 11
...

T. D. Noe, Table of n, a(n) for n = 2..3119 (first 99 nonempty rows of triangle, flattened)
Eric Weisstein's World of Mathematics, Primitive Root.

Cf. A001918, A046144 (row lengths), A046145, A046146.

Cf. A060749, A306252 (1st column), A306253 (last/maximum element)

f:= proc(n) local p,k,m,R;
     p:= numtheory:-primroot(n);
     if p = FAIL then return NULL fi;
     m:= numtheory:-phi(n);
     k:= select(i -> igcd(i,m) = 1, [$1..m-1]);
     op(sort(map(t -> p&^t mod n, k)))
end proc:
f(2):= 1:
map(f, [$2..50]); # Robert Israel, Apr 28 2017

a[n_] := Select[Range[n-1], GCD[#, n] == 1 && MultiplicativeOrder[#, n] == EulerPhi[n]& ]; Table[a[n], {n, 1, 30}] // Flatten (* Jean-François Alcover, Oct 23 2012 *)
PrimitiveRootList[Range[Prime[10]]]//Flatten (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 10 2016 *)

a_row(r) = my(v=[], phi=eulerphi(r)); for(i=1, r-1, if(1 == gcd(r, i) && phi == znorder(Mod(i, r)), v=concat(v, i))); v \\ Ruud H.G. van Tol, Oct 23 2023

Edited by Robert Israel, Apr 28 2017

A123475 Product of the primitive roots of prime(n).

Original entry on oeis.org

1, 2, 6, 15, 672, 924, 11642400, 163800, 109681110000, 5590307923200, 970377408, 134088514560000, 138960660963091968000, 874927557504000, 3456156426256013065185600000000, 30688148115024695887527936000000

Offset: 1

T. D. Noe, Sep 27 2006

nonn

Except for n=2, we have a(n)=1 (mod prime(n)).

			a(5)=672 because the primitive roots of 11 are {2,6,7,8}.

C. F. Gauss, Disquisitiones Arithmeticae, Yale, 1965; see p. 52.

Charles R Greathouse IV, Table of n, a(n) for n = 1..145

Cf. A060749 (primitive roots of prime(n)), A088144 (sum of primitive roots of prime(n)).

PrimRoots[p_] := Select[Range[p-1], MultiplicativeOrder[ #,p]==p-1&]; Table[Times@@PrimRoots[Prime[n]], {n,20}]
Times@@@Table[PrimitiveRootList[Prime[n]], {n, 20}] (* Harlan J. Brothers, Sep 02 2023 *)

vecprod(v)=prod(i=1,#v,v[i])
a(n,p=prime(n))=vecprod(select(n->znorder(Mod(n,p))==p-1,[2..p-1]))
apply(p->a(0,p), primes(20)) \\ Charles R Greathouse IV, May 15 2015

use ntheory ":all"; sub list { my $n=shift; grep { znorder($,$n) == $n-1 } 2..$n-1; } say vecprod(list($)) for @{primes(nth_prime(20))}; # Dana Jacobsen, May 15 2015

A138305 Irregular triangle of prime primitive roots of prime(n).

Original entry on oeis.org

2, 2, 3, 3, 5, 2, 7, 2, 7, 11, 3, 5, 7, 11, 2, 3, 13, 5, 7, 11, 17, 19, 2, 3, 11, 19, 3, 11, 13, 17, 2, 5, 13, 17, 19, 7, 11, 13, 17, 19, 29, 3, 5, 19, 29, 5, 11, 13, 19, 23, 29, 31, 41, 43, 2, 3, 5, 19, 31, 41, 2, 11, 13, 23, 31, 37, 43, 47, 2, 7, 17, 31, 43, 59, 2, 7, 11, 13, 31, 41, 61, 7

Offset: 2

T. D. Noe, Mar 14 2008

The length of row n is A138304(n).

			2;
2,3;
3,5;
2,7;
2,7,11;
3,5,7,11;

T. D. Noe, Rows n=2..100 of triangle, flattened

Cf. A002233 (least prime primitive root), A060749 (triangle of primitive roots of primes).

Flatten[Table[p=Prime[n]; Select[Prime[Range[n-1]], MultiplicativeOrder[ #,p]==p-1&], {n,100}]]

A002199 Least negative primitive root of n-th prime.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 3, 4, 2, 2, 7, 2, 6, 9, 2, 2, 3, 2, 4, 2, 5, 2, 3, 3, 5, 2, 2, 3, 6, 3, 9, 3, 3, 4, 2, 5, 5, 4, 2, 2, 3, 2, 2, 5, 2, 2, 4, 9, 3, 6, 3, 2, 7, 3, 3, 2, 2, 2, 5, 3, 6, 2, 7, 2, 10, 2, 5, 10, 3, 2, 3, 2, 2, 2, 4, 2, 2, 5, 3, 21, 3, 2, 5, 5, 5, 3, 3, 13, 2, 2, 3, 2, 2, 4, 5, 2, 2, 3, 4, 2, 4, 2, 3

Offset: 1

N. J. A. Sloane

nonn

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.
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=1..10000
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].
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]

Cf. A060749, A001918.

Table[(k=-1;While[MultiplicativeOrder[k,p]!=p-1,k--];-k),{p,Prime@Range@100}] (* Giorgos Kalogeropoulos, Sep 28 2023 *)

a(n) = prime(n) - A071894(n). - T. D. Noe, Oct 24 2005

A266987 Primes p for which the average of the primitive roots equals p/2.

Original entry on oeis.org

2, 5, 13, 17, 19, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 307, 313, 317, 337, 349, 353, 373, 389, 397, 401, 409, 421, 433, 449

Offset: 1

Dimitri Papadopoulos, Jan 08 2016

nonn

From Robert Israel, Feb 01 2016: (Start)
Union of A002144 and A267010.
Contains A002144 because for each of these primes, x is a primitive root iff p-x is a primitive root. (End)

			a(13) = 13 since the primitive roots of 13 are 2, 6, 7, and 11 and the average of these primitive roots is (2+6+7+11)/phi(12) = 26/4 = 13/2.

Robert Israel, Table of n, a(n) for n = 1..8213

Cf. A002144, A008330, A060749, A088144, A266986, A267010.

proots := proc(n)
    local r,eulphi,m;
    if n = 1 then
        return {0} ;
    end if;
    eulphi := numtheory[phi](n) ;
    r := {} ;
    for m from 0 to n-1 do
        if numtheory[order](m,n) = eulphi then
            r := r union {m} ;
        end if;
    end do:
    return r;
end proc:
isA266987 := proc(n)
    local r;
    if isprime(n) then
        r := convert(proots(n),list) ;
        2*add(pr, pr=r)  = n*nops(r) ;
    else
        false;
    end if;
end proc:
for n from 1 to 500 do
    if isA266987(n) then
        printf("%d,",n);
    end if;
end do: # R. J. Mathar, Jan 12 2016
Filter:= proc(p) local x, s,js;
  if p mod 4 = 1 then return true fi;
  x:= numtheory:-primroot(p);
  js:= select(t -> igcd(t,p-1)=1, [$1..p-2]);
  s:= add(x&^ j mod p, j=js);
  evalb(s = p/2 * nops(js))
end proc:
select(Filter, [seq(ithprime(i),i=1..1000)]); # Robert Israel, Feb 01 2016

A = Table[Total[Flatten[Position[Table[MultiplicativeOrder[i, Prime[k]], {i, Prime[k] - 1}],Prime[k] - 1]]]/(EulerPhi[Prime[k] - 1] Prime[k]/2), {k, 1, 100}]; Prime[Flatten[Position[A, _?(# == 1 &)]]]
(* second program (version >= 10): *)
Select[Prime[Range[100]], Mean[PrimitiveRootList[#]] == #/2&] (* Jean-François Alcover, Jan 12 2016 *)

a(n) = prime(A266986(n)).

cp's OEIS Frontend

A001918 Least positive primitive root of n-th prime.

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A088144 Sum of primitive roots of n-th prime.

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A216838 Odd primes for which 2 is not a primitive root.

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A143548 Irregular triangle of numbers k < p^2 such that p^2 divides k^(p-1)-1, with p=prime(n).

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A071894 Largest positive primitive root (

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A046147 Triangle read by rows in which row n lists the primitive roots mod n (omitting numbers n without a primitive root).

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A123475 Product of the primitive roots of prime(n).

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