A019338 -id:A019338 - cp's OEIS frontend

A211244 Order of 8 mod n-th prime: least k such that prime(n) divides 8^k-1.

0, 2, 4, 1, 10, 4, 8, 6, 11, 28, 5, 12, 20, 14, 23, 52, 58, 20, 22, 35, 3, 13, 82, 11, 16, 100, 17, 106, 12, 28, 7, 130, 68, 46, 148, 5, 52, 54, 83, 172, 178, 60, 95, 32, 196, 33, 70, 37, 226, 76, 29, 119, 8, 50, 16, 131, 268, 45, 92, 70, 94, 292, 34, 155, 52

Offset: 1

T. D. Noe, Apr 11 2012

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

Cf. A053451 (order of 8 mod 2n+1), A019338 (full reptend primes in base 8).

In other bases: A014664, A062117, A082654, A211241, A211242, A211243, A211245, A002371, A372801.

A000040:=Filtered([1..350],IsPrime);;
List([1..Length(A000040)],n->OrderMod(8,A000040[n])); # Muniru A Asiru, Feb 06 2019

nn = 8; Table[If[Mod[nn, p] == 0, 0, MultiplicativeOrder[nn, p]], {p, Prime[Range[100]]}]

a(n,{base=8}) = my(p=prime(n)); if(base%p, znorder(Mod(base,p)), 0) \\ Jianing Song, May 13 2024

a(n) = A014664(n)/gcd(3, A014664(n)). - Jianing Song, May 13 2024

A167796 Numbers with primitive root 8.

Original entry on oeis.org

3, 5, 11, 25, 29, 53, 59, 83, 101, 107, 121, 125, 131, 149, 173, 179, 197, 227, 269, 293, 317, 347, 389, 419, 443, 461, 467, 491, 509, 557, 563, 587, 625, 653, 659, 677, 701, 773, 797, 821, 827, 841, 941, 947, 1019, 1061, 1091, 1109, 1187, 1229, 1259, 1277

Offset: 1

T. D. Noe, Nov 12 2009

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A019338 (primes with primitive root 8)

pr=8; Select[Range[2,2000], MultiplicativeOrder[pr,# ] == EulerPhi[ # ] &]

is(n)=if(n%2==0, return(0)); my(p=eulerphi(n)); znorder(Mod(8, n), p)==p \\ Charles R Greathouse IV, Jan 04 2025

A105874 Primes for which -2 is a primitive root.

Original entry on oeis.org

5, 7, 13, 23, 29, 37, 47, 53, 61, 71, 79, 101, 103, 149, 167, 173, 181, 191, 197, 199, 239, 263, 269, 271, 293, 311, 317, 349, 359, 367, 373, 383, 389, 421, 461, 463, 479, 487, 503, 509, 541, 557, 599, 607, 613, 647, 653, 661, 677, 701, 709, 719, 743, 751, 757, 773, 797

Offset: 1

N. J. A. Sloane, Apr 24 2005

nonn

Also primes for which (p-1)/2 (==-1/2 mod p) is a primitive root. [Joerg Arndt, Jun 27 2011]

Joerg Arndt, Table of n, a(n) for n = 1..10000
L. J. Goldstein, Density questions in algebraic number theory, Amer. Math. Monthly, 78 (1971), 342-349.
Index entries for primes by primitive root

Cf. A001122, A019334-A019338, A001913, A019339-A019367 etc., A105875-A105914.

with(numtheory); f:=proc(n) local t1,i,p; t1:=[]; for i from 1 to 500 do p:=ithprime(i); if order(n,p) = p-1 then t1:=[op(t1),p]; fi; od; t1; end; f(-2);

pr=-2; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &] (* N. J. A. Sloane, Jun 01 2010 *)
a[p_,q_]:=Sum[2 Cos[2^n Pi/((2 q+1) (2 p+1))], {n,1,2 q p}];
Select[Range[400], Reduce[a[#, 1] == 1, Integers] &];
2 % + 1 (* Gerry Martens, Apr 28 2015 *)

forprime(p=3,10^4,if(p-1==znorder(Mod(-2,p)),print1(p", "))); /* Joerg Arndt, Jun 27 2011 */

from sympy import n_order, nextprime
from itertools import islice
def A105874_gen(startvalue=3): # generator of terms >= startvalue
    p = max(startvalue-1,2)
    while (p:=nextprime(p)):
        if n_order(-2,p) == p-1:
            yield p
A105874_list = list(islice(A105874_gen(),20)) # Chai Wah Wu, Aug 11 2023

Let a(p,q)=sum(n=1,2*p*q,2*cos(2^n*Pi/((2*q+1)*(2*p+1)))). Then 2*p+1 is a prime belonging to this sequence when a(p,1)==1. - Gerry Martens, May 21 2015

A084865 Primes of the form 2x^2 + 3y^2.

Original entry on oeis.org

2, 3, 5, 11, 29, 53, 59, 83, 101, 107, 131, 149, 173, 179, 197, 227, 251, 269, 293, 317, 347, 389, 419, 443, 461, 467, 491, 509, 557, 563, 587, 653, 659, 677, 683, 701, 773, 797, 821, 827, 941, 947, 971, 1013, 1019, 1061, 1091, 1109, 1163, 1181, 1187

Offset: 1

Reinhard Zumkeller, Jun 10 2003

Subsequence of A084864; A084863(a(n))>0.
Conjecture: A084863(a(n))=1?
Is it true that a(n) = A019338(n+1)?
Comment: The truth of the conjecture A084863(a(n))=1 follows from the genus theory of quadratic forms (see Cox, page 61). By comparing enough terms, we see that the conjecture a(n) = A019338(n+1) is false. - T. D. Noe, May 02 2008
Appears to be the primes p such that (p mod 6)*(Fibonacci(p) mod 6)=25. - Gary Detlefs, May 26 2014

			A000040(17) = 59 = 32 + 27 = 2*4^2 + 3*3^2, therefore 59 is a term.

David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A084863, A084864, A019338, A084866, A139827.

Primes in A002480.

QuadPrimes2[2, 0, 3, 10000] (* see A106856 *)

list(lim)=my(v=List(),w,t); for(x=0, sqrtint(lim\2), w=2*x^2; for(y=0, sqrtint((lim-w)\3), if(isprime(t=w+3*y^2), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 09 2017

The primes are congruent to {2, 3, 5, 11} (mod 24). - T. D. Noe, May 02 2008

A307627 Primes p such that 2 is a primitive root modulo p while 8 is not.

Original entry on oeis.org

13, 19, 37, 61, 67, 139, 163, 181, 211, 349, 373, 379, 421, 523, 541, 547, 613, 619, 661, 709, 757, 787, 829, 853, 859, 877, 883, 907, 1117, 1123, 1171, 1213, 1237, 1291, 1381, 1453, 1483, 1531, 1549, 1621, 1669, 1693, 1741, 1747, 1861, 1867, 1987, 2029, 2053

Offset: 1

Jianing Song, Apr 19 2019

nonn

Primes p such that 2 is a primitive root modulo p (i.e., p is in A001122) and that p == 1 (mod 3).

According to Artin's conjecture, the number of terms <= N is roughly ((2/5)*C)*PrimePi(N), where C is the Artin's constant = A005596, PrimePi = A000720. Compare: the number of terms of A001122 that are no greater than N is roughly C*PrimePi(N).

			For p = 67, the multiplicative order of 2 modulo 67 is 66, while 8^22 == 2^(3*22) == 1 (mod 67), so 67 is a term.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Artin's constant
Wikipedia, Artin's conjecture on primitive roots

Complement of A019338 with respect to A001122.

Cf. also A005596, A000720, A307628.

select(p -> isprime(p) and numtheory:-order(2,p) = p-1,
[seq(i,i=1..10000,6)]); # Robert Israel, Apr 23 2019

Select[Prime@ Range[5, 310], And[FreeQ[#, 8], ! FreeQ[#, 2]] &@ PrimitiveRootList@ # &] (* Michael De Vlieger, Apr 23 2019 *)

forprime(p=3, 2000, if(znorder(Mod(2, p))==(p-1) && p%3==1, print1(p, ", ")))

A231371 Squarefree composite numbers k such that 8 is a primitive root for all prime factors of k.

Original entry on oeis.org

15, 33, 55, 87, 145, 159, 165, 177, 249, 265, 295, 303, 319, 321, 393, 415, 435, 447, 505, 519, 535, 537, 583, 591, 649, 655, 681, 745, 795, 807, 865, 879, 885, 895, 913, 951, 957, 985, 1041, 1111, 1135, 1167, 1177, 1245, 1257, 1329, 1345, 1383, 1401, 1441

Offset: 1

Arkadiusz Wesolowski, Nov 08 2013

nonn

If k is the smallest integer satisfying 10^k == 1 (mod p), we say that 10 has order k (mod p). If n is the product of distinct primes p_i, the period of 1/n in base b is the least common multiple of the orders of b (mod p_i), provided b and n are relatively prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Primitive Root.
Wikipedia, Octal.

Subsequence of A024556.

Cf. A019338, A231370, A231372.

q[n_] := CompositeQ[n] && SquareFreeQ[n] && AllTrue[FactorInteger[n][[;;,1]],  MultiplicativeOrder[8, #] == # - 1 &]; Select[Range[1441], q] (* Amiram Eldar, Oct 03 2021 *)

cp's OEIS Frontend

A211244 Order of 8 mod n-th prime: least k such that prime(n) divides 8^k-1.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Mathematica

PARI

Formula

A167796 Numbers with primitive root 8.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A105874 Primes for which -2 is a primitive root.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A084865 Primes of the form 2x^2 + 3y^2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A307627 Primes p such that 2 is a primitive root modulo p while 8 is not.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A231371 Squarefree composite numbers k such that 8 is a primitive root for all prime factors of k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica