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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 and 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], #

= 2&]&]; Select[Prime[Range[2, 500]], selQ] (* Jean-François 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, ", "))); (Python) from functools import cache from math import gcd from itertools import count, islice from sympy import factorint, prime, n_order @cache def is_squareful(n): return n == 1 or min(factorint(n).values()) > 1 def A268629_gen(): # generator of terms for n in count(1): p = prime(n) for i in range(1, p): if gcd(i, p) == 1 and is_squareful(i) and n_order(i, p)==p-1: break else: yield p A268629_list = list(islice(A268629_gen(), 20)) # Chai Wah Wu, Sep 14 2022 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 June 9 07:14 EDT 2023. Contains 363168 sequences. (Running on oeis4.)