A001918 -id:A001918 - 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 *)

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

Original entry on oeis.org

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 *)

A066529 a(n) is the least index such that the least primitive root of the a(n)-th prime is n, or zero if no such prime exists.

Original entry on oeis.org

1, 2, 4, 0, 9, 13, 20, 0, 0, 65, 117, 566, 88, 173, 85, 0, 64, 5426, 43, 10217, 80, 474, 326, 44110, 0, 1479, 0, 12443, 1842, 11662, 775, 0, 23559, 5029, 6461, 0, 3894, 5629, 15177, 105242, 14401, 182683, 9204, 7103, 5518399, 23888, 24092, 42304997, 0, 1455704, 27848, 12107, 14837, 205691645, 38451, 12102037, 39370, 28902, 57481, 56379, 90901, 53468, 5918705, 0, 732055, 1738826, 242495, 265666, 10523, 388487, 260680

Offset: 1

Wouter Meeussen, Jan 06 2002

nonn

The corresponding primes are in A023048.

For n < 150, only a(108) is presently unknown. - Robert G. Wilson v, Jan 03 2006

			a(6) = 13 because Prime[13] = 41 is the least prime with least primitive root = 6

Tomás Oliveira e Silva, Least prime primitive root of prime numbers
E. Weisstein, Primitive Roots
Index entries for primes by primitive root

Cf. A001918, A001122, A001123, A023048, A001597.

big = Table[ p = Prime[ n ]; PrimitiveRoot[ p ], {n, 1, 1000000} ]; Flatten[ Table[ Position[ big, n, 1, 1 ]/.{}-> 0, {n, 79} ] ] (* First load package NumberTheory`NumberTheoryFunctions` *)
(* first load package *) << NumberTheory`NumberTheoryFunctions` (* then do *) t = Table[0, {100}]; Do[a = PrimitiveRoot@Prime@n; If[a < 101 && t[[a]] == 0, t[[a]] = n], {n, 10^6}]; t (* Robert G. Wilson v, Dec 15 2005 *)

a(n) = 0 iff n is a perfect power (A001597) > 1. - Robert G. Wilson v, Jan 03 2006

a(n) = min { k | A001918(k) = n } U {0} = A000720(A023048(n)) (or zero). - M. F. Hasler, Jun 01 2018

Edited by Dean Hickerson, Jan 14 2002

Further terms from Robert G. Wilson v, Jan 03 2006

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

A002199 Least negative primitive root of n-th prime.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 3, 4, 2, 2, 7, 2, 6, 9, 2, 2, 3, 2, 4, 2, 5, 2, 3, 3, 5, 2, 2, 3, 6, 3, 9, 3, 3, 4, 2, 5, 5, 4, 2, 2, 3, 2, 2, 5, 2, 2, 4, 9, 3, 6, 3, 2, 7, 3, 3, 2, 2, 2, 5, 3, 6, 2, 7, 2, 10, 2, 5, 10, 3, 2, 3, 2, 2, 2, 4, 2, 2, 5, 3, 21, 3, 2, 5, 5, 5, 3, 3, 13, 2, 2, 3, 2, 2, 4, 5, 2, 2, 3, 4, 2, 4, 2, 3

Offset: 1

N. J. A. Sloane

nonn

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 864.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
A. E. Western and J. C. P. Miller, Tables of Indices and Primitive Roots, Royal Society Mathematical Tables, Vol. 9, Cambridge Univ. Press, 1968 [Annotated scans of selected pages]

Cf. A060749, A001918.

Table[(k=-1;While[MultiplicativeOrder[k,p]!=p-1,k--];-k),{p,Prime@Range@100}] (* Giorgos Kalogeropoulos, Sep 28 2023 *)

a(n) = prime(n) - A071894(n). - T. D. Noe, Oct 24 2005

A047934 Consider primes p with least positive primitive root g such that q=p+g is next prime after p; sequence gives values of p.

Original entry on oeis.org

2, 3, 5, 11, 29, 59, 101, 107, 149, 151, 179, 197, 227, 251, 269, 271, 337, 347, 367, 419, 461, 659, 733, 821, 827, 971, 991, 1019, 1021, 1061, 1091, 1229, 1277, 1301, 1427, 1451, 1619, 1667, 1787, 1877, 1931, 1949, 1997, 2027, 2141, 2237, 2267, 2309

Offset: 1

Felice Russo

			11 has primitive root 2 and 11+2 = 13 is prime after 11, so 11 is in sequence.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 864.

T. D. Noe, Table of n, a(n) for n=1..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Index entries for primes by primitive root

Cf. A047933, A047935. See also A001918.

ok[p_] := (p + PrimitiveRoot[p] == NextPrime[p]); Select[Prime[Range[343]], ok]  (* Jean-François Alcover, May 03 2011 *)
Transpose[Select[Partition[Prime[Range[400]],2,1],#[[2]]-#[[1]] == PrimitiveRoot[ #[[1]]]&]][[1]] (* Harvey P. Dale, Oct 08 2012 *)

More terms from James Sellers, Dec 22 1999

A084739 Let p = n-th prime, then a(n) = smallest prime having p as its least prime primitive root.

Original entry on oeis.org

3, 7, 23, 41, 109, 457, 311, 191, 2137, 409, 1021, 1031, 1811, 271, 14293, 2791, 55441, 35911, 57991, 221101, 23911, 11971, 110881, 103091, 71761, 513991, 290041, 31771, 448141, 2447761, 674701, 3248701, 2831011, 690541, 190321, 2080597, 4076641

Offset: 1

N. J. A. Sloane, Jul 03 2003

Smallest prime(m) such that A002233(m) = prime(n). (Corrected by Jonathan Sondow, Feb 02 2013)

N. J. A. Sloane, Table of n, a(n) for n=1..97 (from the web page of Tomás Oliveira e Silva)
Tomás Oliveira e Silva, Least primitive root of prime numbers
A. Paszkiewicz and A. Schinzel, On the least prime primitive root modulo a prime, Math. Comp. 71 (2002), no. 239, 1307-1321.

Cf. A084735, A001918, A079061. For records see A133434.

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com) and Don Reble, Jul 03 2003

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

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A066529 a(n) is the least index such that the least primitive root of the a(n)-th prime is n, or zero if no such prime exists.

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

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A002199 Least negative primitive root of n-th prime.

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A047934 Consider primes p with least positive primitive root g such that q=p+g is next prime after p; sequence gives values of p.

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A084739 Let p = n-th prime, then a(n) = smallest prime having p as its least prime primitive root.

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

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