A103310 -id:A103310 - cp's OEIS frontend

A103309 Smallest prime primitive root of n that is less than n, or 0 if none exists.

0, 0, 0, 2, 3, 2, 5, 3, 0, 2, 3, 2, 0, 2, 3, 0, 0, 3, 5, 2, 0, 0, 7, 5, 0, 2, 7, 2, 0, 2, 0, 3, 0, 0, 3, 0, 0, 2, 3, 0, 0, 7, 0, 3, 0, 0, 5, 5, 0, 3, 3, 0, 0, 2, 5, 0, 0, 0, 3, 2, 0, 2, 3, 0, 0, 0, 0, 2, 0, 0, 0, 7, 0, 5, 5, 0, 0, 0, 0, 3, 0, 2, 7, 2, 0, 0, 3, 0, 0, 3, 0, 0, 0, 0, 5, 0, 0, 5, 3, 0, 0, 2, 0, 5, 0

Offset: 0

Harry J. Smith, Jan 29 2005

Differs from A046145 only for indices n = 2, 41, 109, 151, 229, ...; see A103335. - Jeppe Stig Nielsen, Mar 06 2020

Robert Israel, Table of n, a(n) for n = 0..10000
G. Martin, The Least Prime Primitive Root and the Shifted Sieve, Acta Arith. 80 (1997), no. 3, 277-288; arXiv:math/9807104 [math.NT], 1998.
Eric Weisstein's World of Mathematics, Primitive Root.

Cf. A001918, A046144, A046145, A046146, A103310, A103335.

F:= proc(n)
  local r;
  r:= numtheory:-primroot(n);
  while r::integer and not isprime(r) do
    r:= numtheory:-primroot(r,n);
  od:
  if r = FAIL then 0 else r fi
end proc:
seq(F(n),n=0..200); # Robert Israel, May 18 2015

a[n_] := SelectFirst[PrimitiveRootList[n], PrimeQ[#] && # < n&] /. Missing["NotFound"] -> 0;
Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Nov 15 2017 *)

A103335 Numbers whose smallest primitive root (A046145) is not prime.

Original entry on oeis.org

1, 2, 41, 109, 151, 229, 251, 271, 313, 337, 362, 367, 409, 439, 542, 626, 674, 733, 761, 818, 878, 971, 991, 1021, 1031, 1069, 1289, 1297, 1303, 1429, 1471, 1489, 1681, 1759, 1783, 1789, 1811, 1871, 1873, 1879, 2062, 2137, 2342, 2411, 2441, 2551, 2594

Offset: 1

Harry J. Smith, Jan 31 2005

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Primitive Root.

Cf. A001918, A046144, A046145, A046146, A103309, A103310, A103336, A103337, A103338.

filter:= proc(n)
  local r;
  r:= numtheory:-primroot(n);
  r <> FAIL and not isprime(r)
end proc:
filter(1):= true:
select(filter, [$1..3000]); Robert Israel, Sep 08 2020

L = {}; Do[ If[!PrimeQ[ Min[ Select[ Range[n], CoprimeQ[#, n] && MultiplicativeOrder[#, n] == CarmichaelLambda[n] &]]],
L = Append[L, n]], {n, 1, 3000}]; L (* Jonathan Sondow, May 17 2017 *)

Offset changed by Robert Israel, Sep 08 2020

A103336 Numbers whose largest primitive root (A046146) is not prime.

Original entry on oeis.org

1, 2, 11, 17, 19, 23, 29, 31, 37, 38, 41, 43, 47, 53, 58, 59, 62, 67, 71, 73, 74, 79, 81, 82, 83, 86, 89, 94, 97, 101, 107, 113, 118, 121, 122, 125, 127, 131, 134, 137, 139, 146, 149, 151, 157, 158, 162, 163, 167, 173, 178, 179, 191, 193, 194, 197, 211, 218, 223

Offset: 1

Harry J. Smith, Jan 31 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Primitive Root.

Cf. A001918, A046144, A046145, A046146, A103309, A103310, A103335, A103337, A103338.

Select[Range[250], #==1 || ((p = PrimitiveRootList[#]) != {} && ! PrimeQ[Max @ p]) &] (* Amiram Eldar, Sep 25 2021 *)

Offset corrected by Amiram Eldar, Sep 25 2021

A103337 Smallest primitive root of numbers in sequence A103335.

Original entry on oeis.org

0, 1, 6, 6, 6, 6, 6, 6, 10, 10, 21, 6, 21, 15, 15, 15, 15, 6, 6, 21, 15, 6, 6, 10, 14, 6, 6, 10, 6, 6, 6, 14, 6, 6, 10, 6, 6, 14, 10, 6, 21, 10, 35, 6, 6, 6, 15, 6, 6, 10, 21, 14, 6, 33, 6, 6, 10, 6, 6, 6, 10, 6, 22, 15, 15, 6, 10, 12, 6, 15, 15, 10, 14, 21, 6, 15, 6, 6, 6, 6, 6, 6, 6, 10, 10, 6

Offset: 1

Harry J. Smith, Jan 31 2005

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Primitive Root.

Cf. A001918, A046144, A046145, A046146, A103309, A103310, A103335, A103336, A103338.

# given list A103335 as at that sequence
[0,seq(numtheory:-primroot(A103335[i]),i=2..nops(A103335))]; # Robert Israel, Sep 08 2020

Offset changed by Robert Israel, Sep 08 2020

A103338 Largest primitive root of numbers in sequence A103336.

Original entry on oeis.org

0, 1, 8, 14, 15, 21, 27, 24, 35, 33, 35, 34, 45, 51, 55, 56, 55, 63, 69, 68, 69, 77, 77, 75, 80, 77, 86, 91, 92, 99, 104, 110, 115, 117, 115, 123, 118, 128, 117, 134, 135, 141, 147, 146, 152, 153, 155, 159, 165, 171, 175, 176, 189, 188, 189, 195, 207, 207, 214

Offset: 1

Harry J. Smith, Jan 31 2005

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Primitive Root.

Cf. A001918, A046144, A046145, A046146, A103309, A103310, A103335, A103336, A103337.

f:= proc(n) local m; uses NumberTheory;
  if n::odd then
    if NumberOfPrimeFactors(n,distinct) > 1 then return NULL fi;
  elif n mod 4 = 0 or NumberOfPrimeFactors(n,distinct) > 2 then return NULL
  fi;
  m:= PrimitiveRoot(n, ith=Totient(Totient(n)));
  if isprime(m) then NULL else m fi
end proc:
f(1):= 0:map(f, [$1..300]); # Robert Israel, Dec 01 2024

Offset changed by Robert Israel, Dec 01 2024

cp's OEIS Frontend

A103309 Smallest prime primitive root of n that is less than n, or 0 if none exists.

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A103335 Numbers whose smallest primitive root (A046145) is not prime.

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A103336 Numbers whose largest primitive root (A046146) is not prime.

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A103337 Smallest primitive root of numbers in sequence A103335.

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A103338 Largest primitive root of numbers in sequence A103336.

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