A103309 -id:A103309 - cp's OEIS frontend

A103310 Largest prime primitive root of n that is less than n, or 0 if none exists.

0, 0, 0, 2, 3, 3, 5, 5, 0, 5, 7, 7, 0, 11, 5, 0, 0, 11, 11, 13, 0, 0, 19, 19, 0, 23, 19, 23, 0, 19, 0, 17, 0, 0, 31, 0, 0, 19, 29, 0, 0, 29, 0, 29, 0, 0, 43, 43, 0, 47, 47, 0, 0, 41, 47, 0, 0, 0, 47, 47, 0, 59, 53, 0, 0, 0, 0, 61, 0, 0, 0, 67, 0, 59, 61, 0, 0, 0, 0, 59, 0, 59, 71, 79, 0, 0, 73

Offset: 0

Harry J. Smith, Jan 29 2005

Robert Israel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Primitive Root.

Cf. A001918, A046144, A046145, A046146, A103309.

hasproot:= proc(n)
  if n::odd then nops(numtheory:-factorset(n))=1
  else padic:-ordp(n,2)=1 and nops(numtheory:-factorset(n/2))=1
  fi
end proc;
hasproot(2):= true: hasproot(4):= true:
f:= proc(n) local p,t;
  if not hasproot(n) then return 0 fi;
  t:= numtheory:-phi(n);
  p:= prevprime(n);
  while not numtheory:-order(p,n)=t do
    if p = 2 then return 0 fi;
    p:= prevprime(p);
  od;
  p
end proc:
f(0):= 0: f(1):= 0: f(2):= 0:
map(f, [$0..100]); # Robert Israel, Sep 08 2020

a[n_] := Module[{R = PrimitiveRootList[n], s}, s = Select[R, # < n && PrimeQ[#]&]; If[s == {}, 0, s[[-1]]]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 01 2023 *)

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

A279398 a(n) is the smallest prime primitive root modulo A193305(n).

Original entry on oeis.org

3, 5, 2, 3, 3, 5, 7, 2, 7, 2, 3, 3, 5, 3, 3, 5, 3, 3, 5, 2, 7, 3, 5, 3, 3, 11, 2, 7, 2, 7, 7, 5, 3, 2, 5, 2, 3, 5, 3, 5, 5, 11, 3, 7, 2, 3, 3, 17, 3, 3, 3, 3, 7, 5, 3, 5, 7, 3

Offset: 1

Wolfdieter Lang, Jan 18 2017

nonn

Values taken from A103309 (Robert Israel).

If there should be no prime primitive root for A193305(n) then a(n) = 0.

			n = 1: 2^k (mod 4) is never 1 for k >=1. 3^1 = 3, 3^2 = 3^phi(4) = 9 == 1 (mod 4).

Cf. A103309, A193305.

cp's OEIS Frontend

A103310 Largest 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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A279398 a(n) is the smallest prime primitive root modulo A193305(n).

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