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A268629 Primes p that have no squareful primitive roots less than p. 1
3, 5, 7, 13, 17, 19, 23, 31, 41, 43, 47, 61, 71, 73, 79, 97, 103, 127, 191, 193, 223, 239, 241, 311, 313, 337, 409, 433, 439, 457, 479, 601, 719, 769, 839, 911, 1009, 1031, 1033, 1129, 1151, 1201, 1249, 1319, 1321, 1559, 1801, 2089, 2281, 2521, 2689, 2999, 3049, 3361, 3529, 3889 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

Robert Israel, Table of n, a(n) for n = 1..114

Stephen D. Cohen, Tim Trudgian, On the least square-free primitive root modulo p, arXiv:1602.02440 [math.NT], 2016.

EXAMPLE

The primitive roots of 7 less than 7 are 3 and 5. None of them are squareful so 7 is in the sequence.

8 is a primitive root of 11, and 8 is squareful, so 11 is not in the sequence.

MAPLE

N:= 10^6: # for terms <= N

S:= {1}: p:= 1:

do

  p:= nextprime(p);

  if p^2 > N then break fi;

  S:= S union map(t -> seq(t*p^i, i=2..floor(log[p](N/t))), select(`<=`, S, N/p^2));

od:

S:= sort(convert(S, list)):

nS:= nops(S):

filter:= proc(p) local i;

  if not isprime(p) then return false fi;

  for i from 1 to nS while S[i] < p do

    if numtheory:-order(S[i], p) = p-1 then return false fi

  od;

  true

end proc:

select(filter, [seq(i, i=3..N, 2)]); # Robert Israel, Oct 27 2020

MATHEMATICA

selQ[p_] := NoneTrue[PrimitiveRootList[p], #<p && AllTrue[FactorInteger[#], #[[2]] >= 2&]&];

Select[Prime[Range[2, 500]], selQ] (* Jean-Fran├žois Alcover, Sep 28 2018 *)

PROG

(PARI) ar(p) = my(r, pr, j); r=vector(eulerphi(p-1)); pr=znprimroot(p); for(i=1, p-1, if(gcd(i, p-1)==1, r[j++]=lift(pr^i))); vecsort(r) ; \\ from A060749

isok(p) = {my(v = ar(p)); for (i=1, #v, if (ispowerful(v[i]), return(0)); ); 1; }

lista(nn) = forprime(p=1, nn, if (isok(p), print1(p, ", ")));

CROSSREFS

Cf. A001694, A001918, A060749.

Sequence in context: A154320 A173912 A049231 * A092195 A046066 A327819

Adjacent sequences:  A268626 A268627 A268628 * A268630 A268631 A268632

KEYWORD

nonn

AUTHOR

Michel Marcus, Feb 09 2016

STATUS

approved

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Last modified May 25 00:03 EDT 2022. Contains 354047 sequences. (Running on oeis4.)