A001918 -id:A001918 - cp's OEIS frontend

A254309 Irregular triangular array read by rows: T(n,k) is the least positive primitive root of the n-th prime p=prime(n) raised to successive powers of k (mod p) where 1<=k<=p-1 and gcd(k,p-1)=1.

1, 2, 2, 3, 3, 5, 2, 8, 7, 6, 2, 6, 11, 7, 3, 10, 5, 11, 14, 7, 12, 6, 2, 13, 14, 15, 3, 10, 5, 10, 20, 17, 11, 21, 19, 15, 7, 14, 2, 8, 3, 19, 18, 14, 27, 21, 26, 10, 11, 15, 3, 17, 13, 24, 22, 12, 11, 21, 2, 32, 17, 13, 15, 18, 35, 5, 20, 24, 22, 19

Offset: 1

Geoffrey Critzer, May 03 2015

Each row is a complete set of incongruent primitive roots.
Each row is a permutation of the corresponding row in A060749.
Row lengths are A008330.
T(n,1) = A001918(n).

			1;
2;
2,  3;
3,  5;
2,  8,  7,  6;
2,  6, 11,  7;
3, 10,  5, 11, 14,  7, 12,  6;
2, 13, 14, 15,  3, 10;
5, 10, 20, 17, 11, 21, 19, 15,  7, 14;
2,  8,  3, 19, 18, 14, 27, 21, 26, 10, 11, 15;
Row 6 contains 2,6,11,7 because 13 is the 6th prime number. 2 is the least positive primitive root of 13. The integers relatively prime to 13-1=12 are {1,5,7,11}. So we have: 2^1==2, 2^5==6, 2^7==11, and 2^11==7 (mod 13).

Alois P. Heinz, Rows n = 1..120, flattened

Cf. A001918, A008330, A060749.
Last elements of rows give A255367.
Row sums give A088144.

with(numtheory):
T:= n-> (p-> seq(primroot(p)&^k mod p, k=select(
         h-> igcd(h, p-1)=1, [$1..p-1])))(ithprime(n)):
seq(T(n), n=1..15);  # Alois P. Heinz, May 03 2015

Table[nn = p;Table[Mod[PrimitiveRoot[nn]^k, nn], {k,Select[Range[nn - 1], CoprimeQ[#, nn - 1] &]}], {p,Prime[Range[12]]}] // Grid

A048976 Primes whose least primitive root is an odd prime.

Original entry on oeis.org

7, 17, 23, 31, 43, 47, 71, 73, 79, 89, 97, 103, 113, 127, 137, 157, 167, 191, 193, 199, 223, 233, 239, 241, 257, 263, 277, 281, 283, 307, 311, 331, 353, 359, 383, 397, 401, 431, 433, 449, 457, 463, 479, 487, 499, 503, 521, 569, 571, 577, 593, 599, 601, 607

Offset: 1

Felice Russo

nonn

Original name was "Primes for which an odd prime is a primitive root".

Primes in A317649. - Robert Israel, Aug 03 2018

P. Ribenboim, The new book of prime number records, Springer 1996, pp. 22-25.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for primes by primitive root

Cf. A001918, A317649.

filter:= proc(n) local p;
  p:= numtheory:-primroot(n);
  p>2 and isprime(p)
end proc:
select(filter, [seq(ithprime(i),i=2..200)]); # Robert Israel, Aug 02 2018

Select[Prime@ Range@ 112, If[# == {}, False, And[PrimeQ@ #, # > 2] &@ #[[1]] ] &@ PrimitiveRootList[#] &] (* Michael De Vlieger, Aug 02 2018 *)

More terms from James Sellers, Apr 22 2000

Definition corrected by Robert Israel, Aug 02 2018

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

A070269 Primes for which the smallest positive primitive root is odd and nonprime.

Original entry on oeis.org

2, 409, 439, 3631, 4441, 4657, 8681, 12841, 15889, 16633, 21559, 22751, 28393, 30091, 30937, 32257, 32353, 33811, 33871, 36793, 36871, 41809, 41851, 42193, 46649, 48673, 51631, 55921, 58237, 59053, 59497, 60889, 63691, 64609, 71011

Offset: 1

Benoit Cloitre, May 09 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A047936, A001918.

onpQ[n_]:=Module[{pr=PrimitiveRoot[n,1]},OddQ[pr]&&!PrimeQ[pr]]; Select[ Prime[ Range[7100]],onpQ]  (* Harvey P. Dale, Jun 22 2020 *)

forprime(n=1,100000,if((-1)^(lift(znprimroot(n))*(1-isprime(lift(znprimroot(n)))))==-1,print1(n,",")))

A128895 Least positive primitive root that all of the first n primes share.

Original entry on oeis.org

1, 5, 17, 17, 17, 227, 227, 5297, 5297, 5297, 226817, 1227497, 2270483, 5967617, 8617187, 27311693, 39928787, 39928787, 345664343, 345664343

Offset: 1

Martin Raab, Apr 20 2007

nonn

			A primitive root of 2 must be == 1 (mod 2); for 3, it must be == 2 (mod 3), and for 5 it must be == 2 or 3 (mod 5). The smallest such number is 17, so a(3)=17.

Don Reble, Table of n, a(n) for n = 1..37

Cf. A001918, A060749.

A223036 Primes p whose smallest positive quadratic nonresidue is a primitive root of p.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 107, 113, 127, 131, 137, 139, 149, 163, 167, 173, 179, 181, 193, 197, 199, 211, 223, 227, 233, 239, 241, 257, 263, 269, 281, 293, 317, 347, 349, 353, 359, 373, 379, 383

Offset: 1

Jonathan Sondow, Mar 13 2013

nonn

See the complementary sequence A222717 for comments.

			The smallest positive quadratic nonresidue of 3 is 2, and 2 is a primitive root of 3, so 3 is a member.

Index entries for primes by primitive root

Cf. A001918, A053760, A222717.

nn = 100; NR = (Table[p = Prime[n]; First[ Select[ Range[p], JacobiSymbol[#, p] != 1 &]], {n, nn}]); Select[ Prime[ Range[nn]], MultiplicativeOrder[ NR[[PrimePi[#]]], #] == # - 1 &]

A229899 a(n) = |{0

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 2, 1, 1, 0, 1, 2, 2, 2, 4, 6, 7, 0, 1, 2, 2, 1, 10, 10, 8, 4, 4, 13, 4, 6, 3, 5, 10, 3, 20, 2, 6, 6, 19, 18, 22, 4, 11, 6, 16, 4, 3, 7, 28, 8, 28, 16, 4, 16, 32, 31, 30, 5, 8, 16, 13, 32, 7, 17, 6, 40, 7, 2, 43, 8, 36, 43, 10, 12, 8, 46, 44, 8, 30, 16, 39, 8, 24, 20, 11, 39, 30, 14, 22, 9, 58, 58, 22, 17, 22, 61, 60, 30, 21, 10

Offset: 1

Zhi-Wei Sun, Oct 02 2013

nonn

Conjecture: a(n) > 0 except for n = 1, 2, 3, 4, 6, 10, 18. In other words, for any prime p > 7 not equal to 13 or 29 or 61, there are three consecutive integers which are primitive roots modulo p.
Let p be an odd prime. For any integer c, define S(c,p) to be the sum of the Legendre symbols ((g+c)/p) over all primitive roots g modulo p among 1, ..., p-1. If g is a primitive root modulo p, then so is the inverse g^{-1} of g modulo p, and -((g^{-1}+c)/p) = (g*(g^{-1}+c)/p) = ((1+c*g)/p). So S(1,p) = 0, and also S(-1,p) = 0 when p == 1 (mod 4). The author also showed that S(-c,p) = S(c,p) if p == 1 (mod 4), and that S(c,p) = 0 if p is a Fermat prime and c is a quadratic residue modulo p.
Zhi-Wei Sun also made the following conjectures:
(i) Let p > 13 be a prime not equal to 19 or 31, and let a,b,c be integers with a or c not divisible by p. If p does not divide b^2-4*a*c, then there is a primitive root g modulo p such that a*g^2+b*g+c is a quadratic residue modulo p, and there is also a primitive root h modulo p such that a*h^2+b*h+c is a quadratic nonresidue modulo p.
(ii) Let p be any odd prime, and let a,b,c be integers with a or c not divisible by p. If p does not divide b^2-4*a*c, then the absolute value of the sum of the Legendre symbols ((a*g^2+b*g+c)/p) over all primitive roots g modulo p among 1, ..., p-1 is smaller than 2*sqrt(p).

			a(5) = 1 since 6, 7, 8 are primitive roots modulo p_5 = 11.
a(7) = 2 since 5, 6, 7, 10, 11, 12 are primitive roots modulo p_7 = 17.
a(8) = 1 since 13, 14, 15 are primitive roots modulo p_8 = 19.

Zhi-Wei Sun, Table of n, a(n) for n = 1..400

Cf. A001918.

gp[g_,p_]:=gp[g,p]=Length[Union[Table[Mod[g^k, p],{k,1,p-1}]]]==p-1
a[n_]:=Sum[If[gp[g,Prime[n]]&&gp[g-1,Prime[n]]&&gp[g+1,Prime[n]],1,0],{g,1,Prime[n]-1}]
Table[a[n],{n,1,100}]

A255367 a(n) = r^(p-2) mod p, where p is the n-th prime and r is the least positive primitive root of p.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 6, 10, 14, 15, 21, 19, 7, 29, 19, 27, 30, 31, 34, 61, 44, 53, 42, 30, 39, 51, 62, 54, 91, 38, 85, 66, 46, 70, 75, 126, 63, 82, 67, 87, 90, 91, 181, 116, 99, 133, 106, 149, 114, 191, 78, 205, 69, 42, 86, 158, 135, 226, 111, 94, 189, 147, 123

Offset: 1

Alois P. Heinz, May 04 2015

a(n) is the last element of row n of A254309.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000040, A001918, A254309.

a:= n-> (p-> numtheory[primroot](p)&^(p-2) mod p)(ithprime(n)):
seq(a(n), n=1..70);

a[n_] := With[{p = Prime[n]}, Mod[PrimitiveRoot[p]^(p-2), p]]; Array[a, 70] (* Jean-François Alcover, Mar 24 2017 *)

a(n) = r^(p-2) mod p, with p = A000040(n) and r = A001918(n).

A376008 Primes p such that there exists a cyclic permutation of the nonzero residues modulo p such that v^2 - 4uw == 0 (mod p) for any three consecutive residues u,v,w.

Original entry on oeis.org

3, 17, 251, 257, 433, 641, 1459, 3457, 3889, 21169, 39367, 54001, 65537, 110251, 114689, 139969, 210913, 246241, 274177, 319489, 629857, 746497, 974849, 995329, 1161217, 1299079, 1492993, 1769473, 2020001, 2424833, 2555521, 2654209, 5038849, 5304641, 5419387, 5746001, 6049243, 6561001

Offset: 1

Nikolay Osipov, Dmitry Petukhov, and Max Alekseyev, Sep 05 2024

nonn

In other words, for any three consecutive residues u,v,w, the quadratic polynomial u*x^2 + v*x + w has zero discriminant modulo p.

It is shown that all suitable permutations q for prime p = a(n) can be constructed by starting with q(1) = 1, q(2) = a primitive root modulo p, and then defining q(k) = q(k-1)^2/(4*q(k-2)) mod p for k >= 3. Hence, the number of suitable permutations (up to cyclic rotations) is given by A046144(a(n)).

			For a(2) = 17, a suitable cyclic permutation is (1, 3, 15, 6, 4, 12, 9, 7, 16, 14, 2, 11, 13, 5, 8, 10).

Max Alekseyev, Table of n, a(n) for n = 1..100
N. Osipov et al., Residues on a circle (in Russian), dxdy.ru, 2024.

Contains Fermat primes (A019434) as a subsequence.

Cf. A001918, A002326, A046144, A376010.

forprime(p=3,10^8, s=(p-1)/znorder(Mod(2,p)); if(factor(p-1)[,1]==factor(2*s)[,1] && !(p%4==1 && s%2==1),print1(p,", ")) );

An odd prime p is a term iff for s:=(p-1)/A002326((p-1)/2), radicals of p-1 and 2s coincide, excluding the case p==1 (mod 4) and s==1 (mod 2).

cp's OEIS Frontend

A254309 Irregular triangular array read by rows: T(n,k) is the least positive primitive root of the n-th prime p=prime(n) raised to successive powers of k (mod p) where 1<=k<=p-1 and gcd(k,p-1)=1.

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A048976 Primes whose least primitive root is an odd prime.

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Extensions

A061325 Primes with 12 as smallest positive primitive root.

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A061330 Primes with 18 as smallest positive primitive root.

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A070269 Primes for which the smallest positive primitive root is odd and nonprime.

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PARI

A128895 Least positive primitive root that all of the first n primes share.

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A223036 Primes p whose smallest positive quadratic nonresidue is a primitive root of p.

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A229899 a(n) = |{0

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A255367 a(n) = r^(p-2) mod p, where p is the n-th prime and r is the least positive primitive root of p.

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A376008 Primes p such that there exists a cyclic permutation of the nonzero residues modulo p such that v^2 - 4uw == 0 (mod p) for any three consecutive residues u,v,w.