A306253 -id:A306253 - cp's OEIS frontend

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

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

A306252 Least primitive root mod A033948(n).

Original entry on oeis.org

0, 1, 2, 3, 2, 5, 3, 2, 3, 2, 2, 3, 3, 5, 2, 7, 5, 2, 7, 2, 2, 3, 3, 2, 3, 6, 3, 5, 5, 3, 3, 2, 5, 3, 2, 2, 3, 2, 7, 5, 5, 3, 2, 7, 2, 3, 3, 5, 5, 3, 2, 5, 3, 2, 6, 3, 11, 2, 7, 2, 3, 2, 7, 3, 2, 7, 5, 2, 6, 5, 3, 5, 2, 5, 5, 2, 2, 3, 2, 2, 19, 5, 5, 2, 3, 3, 5

Offset: 1

Charles Paul, Feb 01 2019

nonn

Let U(k) denote the multiplicative group mod k. a(n) = smallest generator for U(A033948(n)). - N. J. A. Sloane, Mar 10 2019

			For n=2, A033948(2) = 2, U(2) is generated by 1.
For n=14, A033948(14) = 18, and U(18) is generated by both 5 and 11; here we select the smallest generator, 5, so a(14) = 5.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A033948 (numbers that have a primitive root), A306253, A081888 (positions of records), A081889 (record values). First column of A046147.

0,op(subs(FAIL=NULL, map(numtheory:-primroot,[$2..1000]))); # Robert Israel, Mar 10 2019

Array[Take[PrimitiveRootList@ #, UpTo[1]] &, 210] // Flatten (* Michael De Vlieger, Feb 02 2019 *)

from math import gcd
roots = [0]
for n in range(2,140):
    # find U(n)
    un = [i for i in range(1,n) if gcd(i,n) == 1]
    # for each element in U(n), check if it's a generator
    order = len(un)
    is_cyclic = False
    for cand in un:
        is_gen = True
        run = 1
        # If it cand^x = 1 for some x < order, it's not a generator
        for _ in range(order-1):
            run = (run * cand) % n
            if run == 1:
                is_gen = False
                break
        if is_gen:
            roots.append(cand)
            is_cyclic = True
            break
print(roots)

More terms from Michael De Vlieger, Feb 02 2019
Edited by N. J. A. Sloane, Mar 10 2019
Edited by Robert Israel, Mar 10 2019

A378592 a(n) is the first number that is the largest primitive root modulo exactly n numbers.

Original entry on oeis.org

4, 1, 3, 47, 5

Offset: 0

Robert Israel, Dec 01 2024

a(n) is the first number that occurs exactly n times in A306253.

Is there any number that occurs more than 4 times in A306253?

			4 is not the largest primitive root mod any number.
1 is the largest primitive root mod 2.
3 is the largest primitive root mod 4 and mod 5.
47 is the largest primitive root mod 49, 50, and 54.
5 is the largest primitive root mod 6, 7, 9, and 14.

Cf. A306253.

f:= proc(b) local x, t;
  t:= numtheory:-phi(b);
  for x from b-1 by -1 do if igcd(x, b) = 1 and numtheory:-order(x, b)=t then return x fi od
end proc:
f(1):= 0:
cands:= select(t -> t=1 or numtheory:-primroot(t) <> FAIL, [$1..1000]):
R:= map(f, cands):
S:= sort(convert(convert(R,set),list)):
V:= Array(0..10): V[0]:= 4:
for s in S do
  v:= numboccur(s,R);
  if  V[v] = 0 then V[v]:= s fi
od:
convert(V,list); # Robert Israel, Dec 01 2024

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A306252 Least primitive root mod A033948(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Extensions

A378592 a(n) is the first number that is the largest primitive root modulo exactly n numbers.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple