A081889 -id:A081889 - cp's OEIS frontend

A306252 Least primitive root mod A033948(n).

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

A081888 Numbers n such that the least positive primitive root of n is larger than the value for all positive numbers smaller than n.

Original entry on oeis.org

1, 3, 4, 6, 22, 118, 191, 362, 842, 2042, 2342, 3622, 16022, 29642, 66602, 110881, 143522, 535802, 5070662, 6252122, 6497402, 10219442, 69069002, 1130187962

Offset: 1

Sven Simon, Mar 30 2003

A081889 gives the primitive roots itself. Difference from A002229, A002230: In consideration of all n having primitive roots. A002229, A002230 only primes.

Cf. A081889, A002229, A002230. Positions of records of A306252.

a306252 := proc(n::integer)
    local r;
    r := numtheory[primroot](n) ;
    if r <> FAIL then
        return r ;
    else
        return -1 ;
    end if;
end proc:
A081888 := proc()
    local rec,n,lpr ;
    rec := -1 ;
    for n from 1 do
        lpr := a306252(n) ;
        if lpr > rec then
            printf("%d,\n",n) ;
            rec := lpr ;
        end if;
    end do:
end proc:
A081888() ; # R. J. Mathar, Apr 04 2019

nmax = 10^5;
r[n_] := r[n] = Module[{prl = PrimitiveRootList[n]}, If[prl == {}, -1, prl[[1]]]]; r[1] = 1;
Reap[Module[{rec = -1, n, lpr}, For[n = 1, n <= nmax, n++, lpr = r[n]; If[lpr > rec, Print[n, " ", lpr]; Sow[n]; rec = lpr]]]][[2, 1]] (* Jean-François Alcover, Jun 19 2023, after R. J. Mathar *)

from sympy import primitive_root
from itertools import count, islice
def f(n): r = primitive_root(n); return r if r != None else 0
def agen(r=0): yield from ((m, r:=f(m))[0] for m in count(1) if f(m) > r)
print(list(islice(agen(), 18))) # Michael S. Branicky, Feb 13 2023

Numbers 1, 2, 4, p^m and 2*p^m have primitive roots for odd primes p and m >=1 natural number.

a(24) from Michael S. Branicky, Feb 20 2023

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A306252 Least primitive root mod A033948(n).

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