cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A241043 Primes having primitive roots 2 and 3.

Original entry on oeis.org

5, 19, 29, 53, 101, 139, 149, 163, 173, 197, 211, 269, 293, 317, 379, 389, 461, 509, 557, 653, 677, 701, 773, 797, 821, 859, 907, 941, 1061, 1109, 1123, 1229, 1277, 1291, 1301, 1373, 1483, 1493, 1637, 1733, 1747, 1901, 1949, 1973, 1987, 1997, 2069, 2083
Offset: 1

Views

Author

T. D. Noe, Apr 16 2014

Keywords

Links

Crossrefs

Cf. A001122, A019334, A213052.

Programs

  • Mathematica
    fQ[p_, n_] := MultiplicativeOrder[p, n] == n - 1; Select[Prime[Range[400]], fQ[2, #] && fQ[3, #] &]

A241044 Primes having primitive roots 2, 3, and 5.

Original entry on oeis.org

53, 173, 197, 293, 317, 557, 653, 677, 773, 797, 907, 1277, 1373, 1483, 1493, 1637, 1733, 1747, 1987, 1997, 2083, 2213, 2237, 2333, 2357, 2467, 2477, 2683, 2693, 2837, 2957, 3307, 3413, 3533, 3547, 3557, 3643, 3677, 3797, 3917, 4003, 4013, 4133, 4157
Offset: 1

Views

Author

T. D. Noe, Apr 16 2014

Keywords

Links

Crossrefs

Cf. A001122, A019334, A019335, A213052.

Programs

  • Mathematica
    fQ[p_, n_] := MultiplicativeOrder[p, n] == n - 1; Select[Prime[Range[600]], fQ[2, #] && fQ[3, #] && fQ[5, #] &]
Select[Prime[Range[600]],SequenceCount[PrimitiveRootList[#],{2,3,5}]>0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 03 2018 *)

A241045 Primes having primitive roots 2, 3, 5, and 7.

Original entry on oeis.org

173, 293, 677, 773, 797, 907, 1277, 1637, 1747, 2083, 2357, 2477, 2693, 2957, 3533, 3797, 4133, 4157, 4373, 4493, 4603, 4637, 4877, 4973, 5333, 5477, 5717, 5813, 5923, 6053, 6173, 6317, 6547, 6653, 6763, 7013, 7517, 8237, 8573, 8693, 8837, 9173, 9533
Offset: 1

Views

Author

T. D. Noe, Apr 16 2014

Keywords

Links

Crossrefs

Cf. A001122, A019334, A019335, A019337, A213052.

Programs

  • Mathematica
    fQ[p_, n_] := MultiplicativeOrder[p, n] == n - 1; Select[Prime[Range[1200]], fQ[2, #] && fQ[3, #] && fQ[5, #] && fQ[7, #] &]
Select[Prime[Range[1200]],SubsetQ[PrimitiveRootList[#],{2,3,5,7}]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 16 2020 *)

A241046 Primes having primitive roots 2, 3, 5, 7, and 11.

Original entry on oeis.org

173, 293, 677, 2083, 2477, 3533, 3797, 4133, 4157, 4373, 4603, 4637, 5477, 5717, 5923, 6173, 7013, 9173, 9533, 9677, 10853, 11587, 12437, 13037, 13397, 13613, 13877, 14717, 14957, 15077, 15413, 16253, 17093, 17573, 17597, 18413, 18773, 18917, 19157, 19997
Offset: 1

Views

Author

T. D. Noe, Apr 16 2014

Keywords

Links

Crossrefs

Cf. A001122, A001123, A001124, A001126, A019339, A213052.

Programs

  • Mathematica
    fQ[p_, n_] := MultiplicativeOrder[p, n] == n - 1; Select[Prime[Range[2300]], fQ[2, #] && fQ[3, #] && fQ[5, #] && fQ[7, #] && fQ[11, #] &]

A241047 Primes having primitive roots 2, 3, 5, 7, 11, and 13.

Original entry on oeis.org

293, 2477, 4373, 6173, 7013, 9173, 9677, 10853, 13037, 13397, 13613, 13877, 14957, 15413, 17093, 17597, 18413, 18917, 19157, 22277, 22613, 24317, 26813, 27653, 27893, 29333, 30197, 31517, 33893, 34613, 34877, 35573, 37253, 40493, 41117, 41333, 42437
Offset: 1

Views

Author

T. D. Noe, Apr 16 2014

Keywords

Links

Crossrefs

Cf. A001122, A019334, A019335, A019337, A019339, A019341, A213052.

Programs

  • Mathematica
    fQ[p_, n_] := MultiplicativeOrder[p, n] == n - 1; Select[Prime[Range[4500]], fQ[2, #] && fQ[3, #] && fQ[5, #] && fQ[7, #] && fQ[11, #] && fQ[13, #] &]

A241048 Primes having primitive roots 2, 3, 5, 7, 11, 13, and 17.

Original entry on oeis.org

2477, 9173, 10853, 13877, 14957, 15413, 22277, 22613, 24317, 27653, 30197, 34877, 37253, 41117, 41333, 42437, 42677, 43973, 48677, 51413, 55733, 61613, 62597, 63773, 66293, 72533, 73757, 74093, 76733, 79397, 79757, 82997, 86357, 90173, 92237, 92333, 95597
Offset: 1

Views

Author

T. D. Noe, Apr 16 2014

Keywords

Links

Crossrefs

Cf. A001122, A019334, A019335, A019337, A019339, A019341, A019344, A213052.

Programs

  • Mathematica
    fQ[p_, n_] := MultiplicativeOrder[p, n] == n - 1; Select[Prime[Range[10000]], fQ[2, #] && fQ[3, #] && fQ[5, #] && fQ[7, #] && fQ[11, #] && fQ[13, #] && fQ[17, #] &]

A350121 Increasing sequence of primes p == 3 (mod 4) such that all of 2,3,5,...,prime(n) are primitive roots mod p.

Original entry on oeis.org

3, 19, 907, 1747, 2083, 101467, 350443, 916507, 1014787, 6603283, 27068563, 45287587, 226432243, 243060283, 3946895803, 5571195667, 9259384843, 19633449763, 229012273627, 965558895907, 2793054173947, 5142304754563
Offset: 1

Views

Author

Paul Vanderveen, Dec 15 2021

Keywords

Comments

It is possible, although rather unlikely, that any primes congruent to 3 (mod 4) will appear in A213052.
a(19) > 10^11.

Examples

			a(2) = 19 since 19 is the smallest prime (congruent to 3 (mod 4)) such that the first two primes (2 and 3) are primitive roots.

Crossrefs

Cf. A213052.

Programs

  • Mathematica
    max=0;Do[n=Prime@i;If[Mod[n,4]==3,k=1;While[MultiplicativeOrder[Prime@k,n]==n-1,k++];If[k-1>max,Print@n;max++]],{i,10^6}] (* Giorgos Kalogeropoulos, Dec 17 2021 *)
  • PARI
    N=10^10;
default(primelimit, N);
A=2;
{ forprime (p=3, N,
    if (p%4==3,
    q = 1;
    forprime (a=2, A,
        if ( znorder(Mod(a, p)) != p-1,  q=0; break() );
    );
    if ( q, A=nextprime(A+1); print1(p, ", ") );
    );
); }

Extensions

a(19) from Daniel Suteu, Dec 20 2021
a(20)-a(21) from Paul Vanderveen, May 08 2025

A351921 a(n) is the smallest nonzero number k such that gcd(prime(1)^k + 1, prime(2)^k + 1, ..., prime(n)^k + 1) > 1 and gcd(prime(1)^k + 1, prime(2)^k + 1, ..., prime(n+1)^k + 1) = 1.

Original entry on oeis.org

2, 26, 21, 86, 33, 1238, 4401, 4586, 16161, 18561, 81, 37046, 85478, 180146, 339866
Offset: 2

Views

Author

Gleb Ivanov, Feb 25 2022

Keywords

Comments

Apparently, a(n) = (A307965(n+1) + 1)/2 - 1 for n>=3. - Hugo Pfoertner, Mar 02 2022

Crossrefs

Cf. A213052, A241043, A307965.

Programs

  • Mathematica
    a[n_] := Module[{k = 1, p = Prime[Range[n + 1]]}, While[GCD @@ (Most[p]^k + 1) == 1 || GCD @@ (p^k + 1) > 1, k++]; k]; Array[a, 10, 2] (* Amiram Eldar, Feb 26 2022 *)
  • PARI
    isok(k, n) = my(v = vector(n+1, i, prime(i)^k+1)); (gcd(v) == 1) && (gcd(Vec(v, n)) != 1);
a(n) = my(k=1); while (!isok(k, n), k++); k; \\ Michel Marcus, Mar 18 2022
  • Python
    from sympy import sieve
from math import gcd
from functools import reduce
sieve.extend_to_no(50)
pr = list(sieve._list)
terms = [0]*100
for i in range(2, 85478+1):
    k,g,len_f = 1,2,0
    while g != 1:
        k += 1
        len_f += 1
        g = reduce(gcd, [t**i + 1 for t in pr[:k]])
    if len_f > 1 and terms[len_f] == 0:
        terms[len_f] = i
print(terms[2:15])

Extensions

a(15)-a(16) from Jon E. Schoenfield, Mar 01 2022

A355016 Least prime p such that the n smallest primitive roots modulo p are the first n primes.

Original entry on oeis.org

3, 5, 53, 173, 2083, 188323, 350443, 350443, 1014787, 29861203, 154363267
Offset: 1

Views

Author

Giorgos Kalogeropoulos, Jul 02 2022

Keywords

Examples

			a(3) = 53 because the primitive roots mod 53 are 2, 3, 5, 8, 12, 14, 18, 19, 20, 21, 22, 26, 27, ... and the 3 smallest are the first 3 primes 2, 3 and 5.

Crossrefs

Cf. A213052, A350121.

Programs

  • Mathematica
    p=2;t=1;Do[k=2;p=NextPrime@p;While[!Xor[PrimeQ@k,MultiplicativeOrder[k,p]==p-1],k++];If[k>Prime@t,Print@p;t++;p=NextPrime[p,-1]],10^6]
Showing 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.