A261438 -id:A261438 - cp's OEIS frontend

A261388 a(n) is the length of the longest stretch of consecutive primitive roots of the multiplicative group modulo prime(n).

1, 1, 2, 1, 3, 2, 3, 3, 3, 2, 3, 4, 3, 3, 4, 5, 5, 2, 3, 3, 3, 3, 7, 6, 5, 4, 5, 6, 4, 3, 4, 4, 5, 4, 6, 4, 4, 4, 6, 5, 6, 3, 5, 4, 5, 3, 4, 5, 7, 4, 7, 6, 4, 5, 6, 7, 9, 4, 4, 4, 9, 5, 4, 5, 4, 6, 4, 3, 8, 6, 7, 8, 5, 5, 4, 8, 5, 3, 5, 7, 8, 6, 6, 4, 4, 6, 9, 5, 4, 4, 11, 11, 5, 5, 5, 8, 7, 5, 6

Offset: 1

Joerg Arndt, Aug 17 2015

nonn

Joerg Arndt, Table of n, a(n) for n = 1..9592 (terms for all primes < 10^5)

Cf. A261438 (primes corresponding to records).

a[n_] := 1 + Max[ Join[{0}, Length/@ Select[ Split@ Differences @ PrimitiveRootList @ Prime @ n, #[[1]] == 1 &]]]; Array[a, 99] (* Giovanni Resta, Aug 17 2015 *)

consec_pr(p)= \\ max number of consecutive primroots
{
    my( v = vector(p-1) );
    my (g = znprimroot(p) );
    for (j=1, p-1,  if (gcd(p-1,j)==1, v[lift(g^j)]=1 ) );
    my ( m=0, t=0 );
    for (j=1, p-1, if ( v[j]==0, t=0 , t+=1; if ( t>m, m=t ); ); );
    return(m);
}
forprime(p=2,10^3, c=consec_pr(p);  print1( c,", " ); );

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

cp's OEIS Frontend

A261388 a(n) is the length of the longest stretch of consecutive primitive roots of the multiplicative group modulo prime(n).

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