A060749 -id:A060749 - cp's OEIS frontend

A251865 Irregular triangle read by rows in which row n lists the maximal-order elements (

0, 1, 2, 3, 2, 3, 5, 3, 5, 3, 5, 7, 2, 5, 3, 7, 2, 6, 7, 8, 5, 7, 11, 2, 6, 7, 11, 3, 5, 2, 7, 8, 13, 3, 5, 11, 13, 3, 5, 6, 7, 10, 11, 12, 14, 5, 11, 2, 3, 10, 13, 14, 15, 3, 7, 13, 17, 2, 5, 10, 11, 17, 19, 7, 13, 17, 19, 5, 7, 10, 11, 14, 15, 17, 19, 20, 21, 5, 7, 11, 13, 17, 19, 23, 2, 3, 8, 12, 13, 17, 22, 23

Offset: 1

Eric Chen, May 20 2015

Conjecture: Triangle contains all nonsquare numbers infinitely many times.
The orders of the numbers in n-th row mod n are equal to A002322(n).
First and last terms of the n-th row are A111076(n) and A247176(n).
Length of the n-th row is A111725(n).
The n-th row is the same as A046147 for n with primitive roots.

			Read by rows:
n     maximal-order elements (<n) mod n
1     0
2     1
3     2
4     3
5     2, 3
6     5
7     3, 5
8     3, 5, 7
9     2, 5
10    3, 7
11    2, 6, 7, 8
12    5, 7, 11
13    2, 6, 7, 11
14    3, 5
15    2, 7, 8, 13
16    3, 5, 11, 13
17    3, 5, 6, 7, 10, 11, 12, 14
18    5, 11
19    2, 3, 10, 13, 14, 15
20    3, 7, 13, 17
etc.

Eric Chen, First 160 rows of triangle, flattened
Eric Chen, First 1000 rows of triangle

Cf. A111076, A247176, A111725, A046147, A046145, A046146, A046144, A060749, A001918, A071894, A008330.

a[n_] := Select[Range[0, n-1], GCD[#, n] == 1 && MultiplicativeOrder[#, n] == CarmichaelLambda[n]& ]; Table[a[n], {n, 1, 36}]

c(n)=lcm((znstar(n))[2])
a(n)=for(k=0,n-1,if(gcd(k, n)==1 && znorder(Mod(k,n))==c(n), print1(k, ",")))
n=1; while(n<37, a(n); n++)

A266988 The indices of primes for which the average of the primitive roots is < p/2.

Original entry on oeis.org

11, 14, 19, 48, 75, 94, 114, 115, 117, 124, 149, 153, 155, 177, 182, 224, 272, 300, 324, 348, 351, 365, 370, 403, 465, 510, 515, 522, 531, 546, 555, 578, 614, 634, 667, 677, 683, 707, 748, 765, 788, 795, 802, 808, 832, 850, 871, 876, 886, 888, 966, 980

Offset: 1

Dimitri Papadopoulos, Jan 08 2016

nonn

These primes are all of the form p==3 (mod 4). (conjecture)

			p(a[1])=p(11)=31. The primitive roots of 31 are 3, 11, 12, 13, 17, 21, 22, and 24.
Their average is (3+11+12+13+17+21+22+24)/phi(30)=123/8<31/2.

Cf. A008330, A060749, A088144, A266989.

A = Table[Total[Flatten[Position[Table[MultiplicativeOrder[i, Prime[k]], {i, Prime[k] - 1}],Prime[k] - 1]]]/(EulerPhi[Prime[k] - 1] Prime[k]/2), {k, 1, 1000}];Flatten[Position[A, _?(# < 1 &)]]

A268397 a(n) is the smallest prime with at least n consecutive primitive roots.

Original entry on oeis.org

2, 5, 11, 37, 53, 83, 83, 269, 269, 467, 467, 1187, 1559, 1559, 1559, 6803, 6803, 6803, 10559, 10559, 10559, 35279, 38639, 38639, 38639, 38639, 38639

Offset: 1

Dimitri Papadopoulos, Feb 03 2016

			a(4)=37. 37 has the primitive roots 2, 5, 13, 15, 17, 18, 19, 20, 22, 24, 32, and 35 of which 17, 18, 19, and 20 are consecutive.

Cf. A060749, A261438 (has "exactly" instead of "at least").

PrimRoot[n_] :=Flatten[Position[Table[MultiplicativeOrder[i, n], {i, n - 1}],n - 1]];t = {};For[targ = 1, targ <= 22, targ++,flag = 0; For[n = 1, n < 1500, n++,prs = PrimRoot[Prime[n]];numprs = EulerPhi[Prime[n] - 1]; If[targ > numprs, ,For[m = 1, m <= numprs + 1 - targ, m++,temp = Take[prs, {m, m + targ - 1}];If[temp[[1]] + targ - 1 == temp[[targ]] && flag == 0,t = Append[t, Prime[n]]; flag = 1];If[flag == 1, Break[]];]; If[flag == 1, Break[]];];If[flag == 1, Break[]];]]; t
Join[{2},Module[{prl=Table[{p,Max[Length/@Select[Split[ Differences[ PrimitiveRootList[ p]]], #[[1]]==1&]]},{p,Prime[Range[1500]]}]},Table[ SelectFirst[ prl, #[[2]]>=k&],{k,20}]][[All, 1]]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 23 2019 *)

More terms from Harvey P. Dale, Aug 23 2019

A268629 Primes p that have no squareful primitive roots less than p.

Original entry on oeis.org

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

Offset: 1

Michel Marcus, Feb 09 2016

nonn

			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.

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.

Cf. A001694, A001918, A060749.

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

selQ[p_] := NoneTrue[PrimitiveRootList[p], #= 2&]&];
Select[Prime[Range[2, 500]], selQ] (* Jean-François Alcover, Sep 28 2018 *)

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, ", ")));

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

A113771 Largest gap between primitive roots of n-th prime.

Original entry on oeis.org

2, 3, 4, 5, 5, 4, 6, 7, 7, 5, 10, 8, 12, 12, 7, 4, 6, 8, 8, 9, 10, 21, 8, 6, 15, 8, 14, 7, 12, 7, 12, 10, 7, 12, 5, 15, 10, 11, 7, 7, 7, 13, 21, 11, 8, 14, 15, 12, 8, 13, 7, 11, 17, 9, 7, 8, 9, 15, 13, 16, 10, 10, 12, 19, 20, 6, 22, 20, 9, 12, 7, 9, 15, 9, 18, 9, 9, 19, 13, 42, 10, 17, 12, 10

Offset: 1

Franklin T. Adams-Watters, Jan 19 2006

			For n=6, p_6=13, the primitive roots are 2,6,7,11, with largest gap 11-7=4. For n=5, p_5=11, the primitive roots are 2,6,7,8, with largest gap 2-8=5 (mod 11).

R. Osborn, Tables of All Primitive Roots of Odd Primes Less Than 1000, Univ. Texas Press, 1961.

Eric Weisstein's World of Mathematics, Primitive Root.

Cf. A060749 (table), A001918 (least).

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A251865 Irregular triangle read by rows in which row n lists the maximal-order elements (

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A266988 The indices of primes for which the average of the primitive roots is < p/2.

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A268397 a(n) is the smallest prime with at least n consecutive primitive roots.

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A268629 Primes p that have no squareful primitive roots less than p.

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A113771 Largest gap between primitive roots of n-th prime.

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