A306253 Largest primitive root mod A033948(n).

0, 1, 2, 3, 3, 5, 5, 5, 7, 8, 11, 5, 14, 11, 15, 19, 21, 23, 19, 23, 27, 24, 31, 35, 33, 35, 34, 43, 45, 47, 47, 51, 47, 55, 56, 59, 55, 63, 69, 68, 69, 77, 77, 75, 80, 77, 86, 91, 92, 89, 99, 101, 103, 104, 103, 110, 115, 117, 115, 123, 118, 128, 117, 134, 135

Offset

1,3

Comments

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

Links

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

Example

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

Maple

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]):
map(f, cands); # Robert Israel, Mar 10 2019

Mathematica

Join[{0}, Last /@ DeleteCases[PrimitiveRootList[Range[1000]], {}]] (* Jean-François Alcover, Jun 18 2020 *)

Prog

def gcd(x, y):
    # Euclid's Algorithm
    while(y):
        x, y = y, x % y
    return x
roots = []
for n in range(2, 140):
    # find U(n)
un = [i for i in range(n, 0, -1) if (gcd(i, n) is 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:", roots

Crossrefs

See A306252 for smallest roots and A033948 for the sequence of numbers that have a primitive root.

Sequence in context: A098567 A086162 A036703 * A117629 A081165 A289749

Adjacent sequences: A306250 A306251 A306252 * A306254 A306255 A306256

Keyword

nonn

Author

Charles Paul, Feb 01 2019

Extensions

Edited by N. J. A. Sloane, Mar 10 2019.

Status

approved