A019336 -id:A019336 - cp's OEIS frontend

A211242 Order of 6 mod n-th prime: least k such that prime(n) divides 6^k-1.

0, 0, 1, 2, 10, 12, 16, 9, 11, 14, 6, 4, 40, 3, 23, 26, 58, 60, 33, 35, 36, 78, 82, 88, 12, 10, 102, 106, 108, 112, 126, 130, 136, 23, 37, 150, 156, 27, 83, 43, 178, 60, 19, 96, 14, 198, 105, 222, 226, 228, 232, 17, 20, 250, 256, 131, 134, 270, 276, 56, 141

Offset: 1

T. D. Noe, Apr 11 2012

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

Cf. A019336 (full reptend primes in base 6).

In other bases: A014664, A062117, A082654, A211241, A211243, A211244, A211245, A002371.

A000040:=Filtered([1..350],IsPrime);;
List([1..Length(A000040)],n->OrderMod(6,A000040[n])); # Muniru A Asiru, Feb 06 2019

A211242 := proc(n)
    if n<= 2 then
        0 ;
    else
        numtheory[order](6,ithprime(n)) ;
    end if;
end proc:
seq(A211242(n),n=1..80) ; # R. J. Mathar, Jul 17 2024

nn = 6; Table[If[Mod[nn, p] == 0, 0, MultiplicativeOrder[nn, p]], {p, Prime[Range[100]]}]

a(n,{base=6}) = my(p=prime(n)); if(base%p, znorder(Mod(base,p)), 0) \\ Jianing Song, May 13 2024

A167794 Numbers with primitive root 6.

Original entry on oeis.org

11, 13, 17, 41, 59, 61, 79, 83, 89, 103, 107, 109, 113, 121, 127, 131, 137, 151, 157, 169, 179, 199, 223, 227, 229, 233, 251, 257, 271, 277, 289, 347, 367, 373, 397, 401, 419, 443, 449, 467, 487, 491, 521, 563, 569, 587, 593, 613, 641, 659, 661, 683, 709, 733

Offset: 1

T. D. Noe, Nov 12 2009

nonn

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

Cf. A019336 (primes with primitive root 6). Subsequence of A033948.

A167794 := proc(n)
    option remember;
    if n =1 then
        11;
    else
        for a from procname(n-1)+1 do
            if numtheory[order](6,a) = numtheory[phi](a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A167794(n),n=1..80) ; # R. J. Mathar, Sep 15 2021

pr=6; Select[Range[2,2000], MultiplicativeOrder[pr,# ] == EulerPhi[ # ] &]

is(n)=if(gcd(n, 6)>1, return(0)); my(p=eulerphi(n)); znorder(Mod(6, n), p)==p \\ Charles R Greathouse IV, Jan 04 2025

A320384 Primes p such that 3/2 is a primitive root modulo p.

Original entry on oeis.org

7, 11, 17, 31, 37, 41, 59, 83, 89, 103, 107, 109, 113, 127, 131, 137, 151, 157, 179, 223, 227, 229, 233, 251, 257, 271, 277, 347, 349, 353, 367, 397, 421, 443, 449, 467, 491, 521, 541, 563, 569, 587, 593, 607, 613, 631, 641, 659, 661, 683, 733, 757, 761, 809, 827, 853, 857, 877, 929, 953, 967, 971, 977, 991

Offset: 1

Jianing Song, Oct 12 2018

nonn

Primes p such that the smallest positive k such that p divides 3^k - 2^k is p - 1.
All terms are congruent to 7, 11, 13, 17 modulo 24. For other primes p, 3/2 is a quadratic residue modulo p.
By Artin's conjecture, this sequence contains 37.395% of all primes, or 74.79% of all primes congruent to 7, 11, 13, 17 modulo 24.

			3/2 == 5 (mod 7), 5 is a primitive root modulo 7, so 7 is a term. Indeed, 7 does not divide 3^2 - 2^2 or 3^3 - 2^3, but it divides 3^6 - 2^6.
3/2 == 7 (mod 11), 7 is a primitive root modulo 11, so 11 is a term. Indeed, 11 does not divide 3^2 - 2^2 or 3^5 - 2^5, but it divides 3^10 - 2^10.
3/2 == 13 (mod 23), 13^11 == 1 (mod 23), so 23 is not a term. Indeed, 23 divides 3^11 - 2^11.

C. Hooley, On Artin's conjecture, J. reine angewandte Math., 225 (1967) 209-220.
Wikipedia, Artin's conjecture on primitive roots
Wikipedia, Primitive root modulo n
Index entries for sequences related to Artin's conjecture
Index entries for primes by primitive root

Cf. A019336, A320383.

forprime(p=5,10^3,if(p-1==znorder(Mod(3/2,p)),print1(p,", "))); \\ Joerg Arndt, Oct 13 2018

cp's OEIS Frontend

A211242 Order of 6 mod n-th prime: least k such that prime(n) divides 6^k-1.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

A167794 Numbers with primitive root 6.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A320384 Primes p such that 3/2 is a primitive root modulo p.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI