A002223 -id:A002223 - cp's OEIS frontend

A002224 Smallest prime p of form p = 8k+1 such that first n primes (p_1=2, ..., p_n) are quadratic residues mod p.

17, 73, 241, 1009, 2689, 8089, 33049, 53881, 87481, 483289, 515761, 1083289, 3818929, 3818929, 9257329, 22000801, 48473881, 48473881, 175244281, 427733329, 427733329, 898716289, 8114538721, 9176747449, 23616331489, 23616331489, 23616331489, 196265095009, 196265095009, 196265095009, 196265095009, 2871842842801, 2871842842801, 2871842842801, 26437680473689

Offset: 1

N. J. A. Sloane

			32^2 = 2 mod 73, 21^2 = 3 mod 73.

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).
A. E. Western and J. C. P. Miller, Tables of Indices and Primitive Roots. Royal Society Mathematical Tables, Vol. 9, Cambridge Univ. Press, 1968, p. XV.

Michael Hortmann, Table of n, a(n) for n = 1..41
D. H. Lehmer, A sieve problem on "pseudo-squares", Math. Tables Other Aids Comp., 8 (1954), 241-242.
D. H. Lehmer, E. Lehmer and D. Shanks, Integer sequences having prescribed quadratic character, Math. Comp., 24 (1970), 433-451.
D. H. Lehmer, E. Lehmer and D. Shanks, Integer sequences having prescribed quadratic character, Math. Comp., 24 (1970), 433-451 [Annotated 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. A002223, A002225, A002226.

f[n_] := Block[{k = 2}, While[JacobiSymbol[n, Prime[k]] == 1, k++ ]; Prime[k]] (* Robert G. Wilson v *)
np[] := While[p = NextPrime[p]; Mod[p, 8] != 1]; p = 2; A002224 = {}; pp = {2}; np[]; While[Length[A002224] < 25, If[Union[JacobiSymbol[#, p] &[pp]] === {1}, AppendTo[pp, NextPrime[Last[pp]]]; Print[p]; AppendTo[A002224, p], np[]]]; A002224 (* Jean-François Alcover, Sep 09 2011 *)

a(n,startAt=17)=my(v=primes(n)); forprime(p=startAt,, if(p%8>1, next); for(i=1,n, if(kronecker(v[i],p)<1, next(2))); return(p)) \\ Charles R Greathouse IV, Jun 26 2017

More terms from Don Reble, Sep 19 2001

More terms from Mike Oakes, Nov 28 2022

A000229 a(n) is the least number m such that the n-th prime is the least quadratic nonresidue modulo m.

Original entry on oeis.org

3, 7, 23, 71, 311, 479, 1559, 5711, 10559, 18191, 31391, 422231, 701399, 366791, 3818929, 9257329, 22000801, 36415991, 48473881, 175244281, 120293879, 427733329, 131486759, 3389934071, 2929911599, 7979490791, 36504256799, 23616331489, 89206899239, 121560956039

Offset: 1

N. J. A. Sloane

Note that a(n) is always a prime q > prime(n).
For n > 1, a(n) = prime(k), where k is the smallest number such that A053760(k) = prime(n).
One could make a case for setting a(1) = 2, but a(1) = 3 seems more in keeping with the spirit of the sequence.
a(n) is the smallest odd prime q such that prime(n)^((q-1)/2) == -1 (mod q) and b^((q-1)/2) == 1 (mod q) for every natural base b < prime(n). - Thomas Ordowski, May 02 2019

			a(2) = 7 because the second prime is 3 and 3 is the least quadratic nonresidue modulo 7, 14, 17, 31, 34, ... and 7 is the least of these.

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).

N. J. A. Sloane, Table of n, a(n) for n = 1..38 (from the web page of Tomás Oliveira e Silva)
H. J. Godwin, On the least quadratic non-residue, Proc. Camb. Phil. Soc., 61 (3) (1965), 671-672.
A. J. Hanson, G. Ortiz, A. Sabry and Y.-T. Tai, Discrete Quantum Theories, arXiv preprint arXiv:1305.3292 [quant-ph], 2013.
A. J. Hanson, G. Ortiz, A. Sabry, Y.-T. Tai, Discrete quantum theories, (a different version). (To appear in J. Phys. A: Math. Theor., 2014).
Tomás Oliveira e Silva, Least primitive root of prime numbers
Hans Salié, Uber die kleinste Primzahl, die eine gegebene Primzahl als kleinsten positiven quadratischen Nichtrest hat, Math. Nachr. 29 (1965) 113-114.
Yu-Tsung Tai, Discrete Quantum Theories and Computing, Ph.D. thesis, Indiana University (2019).

Cf. A020649, A025021, A053760, A307809. For records see A133435.

Differs from A002223, A045535 at 12th term.

leastNonRes[p_] := For[q = 2, True, q = NextPrime[q], If[JacobiSymbol[q, p] != 1, Return[q]]]; a[1] = 3; a[n_] := For[pn = Prime[n]; k = 1, True, k++, an = Prime[k]; If[pn == leastNonRes[an], Print[n, " ", an];  Return[an]]]; Array[a, 20] (* Jean-François Alcover, Nov 28 2015 *)

Definition corrected by Melvin J. Knight (MELVIN.KNIGHT(AT)ITT.COM), Dec 08 2006

Name edited by Thomas Ordowski, May 02 2019

A002233 a(1) = 1; for n > 1, a(n) = least positive prime primitive root of n-th prime.

Original entry on oeis.org

1, 2, 2, 3, 2, 2, 3, 2, 5, 2, 3, 2, 7, 3, 5, 2, 2, 2, 2, 7, 5, 3, 2, 3, 5, 2, 5, 2, 11, 3, 3, 2, 3, 2, 2, 7, 5, 2, 5, 2, 2, 2, 19, 5, 2, 3, 2, 3, 2, 7, 3, 7, 7, 11, 3, 5, 2, 43, 5, 3, 3, 2, 5, 17, 17, 2, 3, 19, 2, 2, 3, 7, 11, 2, 2, 5, 2, 5, 3, 29, 2, 2, 7, 5, 17, 2, 3, 13, 2, 3, 2, 13, 3, 2, 7, 5, 2, 3, 2, 2, 2

Offset: 1

N. J. A. Sloane

According to Section F9 in Guy's book "Unsolved Problems in Number Theory" (Springer, 2004), P. Erdős asked whether for any large prime p there is a prime q < p so that q is a primitive root modulo p. See also the comments on A223942 related to this sequence. - Zhi-Wei Sun, Mar 29 2013

For n >= 2 the Dirichlet characters modulo prime(n), {Chi_{prime n}{(r,m)}, for n >= 1, r=1..(prime(n)-1) and m = 2..prime(n)-1, are determined from those for m = a(n), i.e., Chi_{prime n}(r,a(n)) = exp(2*Pi*I*(r-1)/(prime(n)-1)) and the power sequence S(n) := {a(n)^k (mod prime(n)), k = 1..(prime(n)-2)} by the strong multiplicity of Chi as Chi_{prime n}(r,m) = (Chi_{prime n}(r,a(n)))^{pos(m,S(n))} where S(n){pos(m,S(n))} = m. For m=1 Chi is always 1. For m = prime(n) Chi is always 0. For n=1 (prime 2) the characters are 1, 0 for r = 1 and m = 1, 2, respectively. See the example for a(4) below. - _Wolfdieter Lang, Jan 19 2017

			n=4, a(4) = 3: Dirichlet characters for prime(4) = 7 from Chi_7(r,3) = exp(Pi*I*(r-1)/3) and the power sequence S(4) = [3, 2, 6, 4, 5]. Hence Chi_7(r,2) = Chi_7(r,3)^2 = exp(2*Pi*I*(r-1)/3), Chi_7(r,4) = Chi_7(r,3)^4, Chi_7(r,5) = Chi_7(r,3)^5, Chi_7(r,6) = Chi_7(r,3)^3. Chi_7(r,1) = 1 and Chi_7(r,7) = 0, for r=1..6. This produces the character modulo 7 table. See the Apostol reference, p. 139, with interchanged rows r = 2..6. - _Wolfdieter Lang_, Jan 19 2017

T. M. Apostol, An Introduction to Analytic Number Theory, Springer-Verlag, NY, 1976, 1986, p. 139.
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).
A. E. Western and J. C. P. Miller, Tables of Indices and Primitive Roots. Royal Society Mathematical Tables, Vol. 9, Cambridge Univ. Press, 1968, p. 2.

T. D. Noe, Table of n, a(n) for n = 1..10000
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]

See A122028 (least primitive root that is prime), A001918 (least positive primitive root), A223942.

a[1] = 1; a[n_] := (p = Prime[n]; Select[Range[p], PrimeQ[#] && MultiplicativeOrder[#, p] == EulerPhi[p] &, 1]) // First; Table[a[n], {n, 100}] (* Jean-François Alcover, Mar 30 2011 *)
a[1] = 1; a[n_] := SelectFirst[PrimitiveRootList[Prime[n]], PrimeQ]; Array[a, 101] (* Jean-François Alcover, Sep 28 2016 *)

leastroot(p)=forprime(q=2,p,if(znorder(Mod(q,p))+1==p,return(q)))
a(n)=if(n>1,leastroot(prime(n)),1) \\ Charles R Greathouse IV, Mar 20 2013

a(n) = A122028(n) for n>1. - Jonathan Sondow, May 18 2017

A002225 a(n) is the smallest prime p such that each of the first n primes has three cube roots mod p.

Original entry on oeis.org

31, 307, 643, 5113, 21787, 39199, 360007, 360007, 4775569, 10318249, 10318249, 65139031, 387453811, 913900417, 2278522747, 2741702809, 25147657981, 118748663779, 156776294593, 747206701687, 1151810360731, 1151810360731, 1151810360731

Offset: 1

N. J. A. Sloane

a(n) is the smallest prime p == 1 (mod 3) such that each of the first n primes is a cubic residue mod p. - Robert Israel, Aug 02 2016

			For n = 2, the first two primes 2 and 3 each have three cube roots mod 307: 2 has cube roots 52, 270, 292 and 3 has cube roots 192, 194, 228. - _Robert Israel_, Aug 02 2016

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).
A. E. Western and J. C. P. Miller, Tables of Indices and Primitive Roots. Royal Society Mathematical Tables, Vol. 9, Cambridge Univ. Press, 1968, p. XVI.

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]

Smallest prime p such that each of the first n primes has q q-th roots mod p: A147972 (q=2), this sequence (q=3), A002226 (q=5), A002227 (q=7), A002228 (q=11), A060363 (q=13), A060364 (q=17).
Cf. A002223, A002224.
Subset of A014752. Except for a(1), subset of A014753. Except for a(1) and a(2), subset of A040044.

Primes:= [2]: pp:= 7:
for n from 1 to 12 do
  for p from pp by 6 while
    not(isprime(p) and andmap(t -> t &^ ((p-1)/3) mod p = 1, Primes))
  do od:
  A[n]:= p;
  pp:= p;
  Primes:= [op(Primes), nextprime(Primes[-1])];
od:
seq(A[i],i=1..12); # Robert Israel, Aug 02 2016

(* This naive program being very slow, limit is set to 8 terms *) lim=8; np[] := While[p=NextPrime[p]; Mod[p,3]!=1]; crQ[n_, p_] := Reduce[ 0A002225={}; While[Length[A002225] < lim, If[And @@ (crQ[#,p]& /@ pp), AppendTo[pp, NextPrime[ Last[pp]]]; Print[p]; AppendTo[A002225,p], np[] ] ]; A002225 (* Jean-François Alcover, Sep 09 2011 *)

More terms from Don Reble, Oct 09 2001
Name corrected by Robert Israel, Aug 02 2016
a(18)-a(23) from Sergey Paramonov, Apr 11 2024

A002230 Primes with record values of the least positive primitive root.

Original entry on oeis.org

2, 3, 7, 23, 41, 71, 191, 409, 2161, 5881, 36721, 55441, 71761, 110881, 760321, 5109721, 17551561, 29418841, 33358081, 45024841, 90441961, 184254841, 324013369, 831143041, 1685283601, 6064561441, 7111268641, 9470788801, 28725635761, 108709927561, 386681163961, 1990614824641, 44384069747161, 89637484042681

Offset: 1

N. J. A. Sloane

R. Osborn, Tables of All Primitive Roots of Odd Primes Less Than 1000, Univ. Texas Press, 1961.
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).
A. E. Western and J. C. P. Miller, Tables of Indices and Primitive Roots. Royal Society Mathematical Tables, Vol. 9, Cambridge Univ. Press, 1968, p. XLIV.

Michel Marcus, Table of n, a(n) for n = 1..38 (using McGown and Sorenson).
Stephen D. Cohen, Tomás Oliveira e Silva, and Tim Trudgian, On Grosswald's conjecture on primitive roots, arXiv:1503.04519 [math.NT], 2015.
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
Kevin J. McGown and Jonathan P. Sorenson, Computation of the least primitive root, arXiv:2206.14193 [math.NT], 2022.
Tomás Oliveira e Silva, Least prime primitive root of prime numbers
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]
Index entries for primes by primitive root

Cf. A002229 (for the primitive roots in question).

Records in A023048, indices in A114885.

s = {2}; rm = 1; Do[p = Prime[k]; r = PrimitiveRoot[p]; If[r > rm, Print[p]; AppendTo[s, p]; rm = r], {k, 10^6}]; s (* Jean-François Alcover, Apr 05 2011 *)
DeleteDuplicates[Table[{p,PrimitiveRoot[p,1]},{p,Prime[Range[61100]]}],GreaterEqual[ #1[[2]],#2[[2]]]&][[All,1]] (* The program generates the first 15 terms of the sequence. *) (* Harvey P. Dale, Aug 22 2022 *)

from sympy import isprime, primitive_root
from itertools import count, islice
def f(n): return 0 if not isprime(n) or (r:=primitive_root(n))==None else r
def agen(r=0): yield from ((m, r:=f(m))[0] for m in count(1) if f(m) > r)
print(list(islice(agen(), 15))) # Michael S. Branicky, Feb 13 2023

More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu)

A002228 Smallest prime p such that first n primes (p_1=2, ..., p_n) are 11th power residues mod p.

Original entry on oeis.org

331, 39139, 253243, 4397207, 21587171, 781712537, 781712537, 25467966877, 1304374210679, 4331892405391

Offset: 1

N. J. A. Sloane

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).
A. E. Western and J. C. P. Miller, Tables of Indices and Primitive Roots. Royal Society Mathematical Tables, Vol. 9, Cambridge Univ. Press, 1968, p. XXIV.

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. A002223-A002227, A060363, A060364.

More terms from Don Reble, Oct 12 2001

a(9)-a(10) from Sergey Paramonov, Apr 14 2024

A147971 Indices of the records in the sequence of smallest positive quadratic nonresidues (A053760).

Original entry on oeis.org

1, 4, 9, 20, 64, 92, 246, 752, 1289, 2084, 3383, 31284, 271259, 618525, 1389315, 2228197, 2914847, 6857528, 7457772, 141236709, 366883983, 1034128714, 3690981956, 4965932454, 7863515482, 19824941433, 195348751601, 292557888940, 2296552237422

Offset: 1

Max Alekseyev, Nov 18 2008

nonn

The corresponding primes are listed in A147970.

Cf. A000229, A002223, A002224, A045535, A053760, A133435, A147969, A147970.

Positive integer n is in this sequence iff A053760(k) < A053760(n) for every k

a(20)-a(29) from Charles R Greathouse IV, Apr 06 2012

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

A002226 Smallest prime p such that first n primes (p_1=2, ..., p_n) are quintic residues mod p.

Original entry on oeis.org

151, 431, 6581, 67651, 241981, 2081921, 3395921, 116900011, 650086271, 858613901, 11736494711, 50888057851, 303855349271, 2459339487751, 3167880361091

Offset: 1

N. J. A. Sloane

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).
A. E. Western and J. C. P. Miller, Tables of Indices and Primitive Roots. Royal Society Mathematical Tables, Vol. 9, Cambridge Univ. Press, 1968, p. XXIII.

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. A002223, A002224, A002225, A002227, etc.

More terms from Don Reble, Oct 10 2001

a(13)-a(15) from Sergey Paramonov, Apr 08 2024

A002227 Smallest prime p such that first n primes (p_1=2, ..., p_n) are 7th power residues mod p.

Original entry on oeis.org

631, 5531, 72661, 865957, 2375059, 32353609, 175175603, 945552637, 945552637, 54144188771, 688203780167, 2701344818803

Offset: 1

N. J. A. Sloane

a(13) > 2^42. - Sergey Paramonov, Apr 12 2024

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. E. Western and J. C. P. Miller, Tables of Indices and Primitive Roots. Royal Society Mathematical Tables, Vol. 9, Cambridge Univ. Press, 1968, p. XXIII.

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. A002223, A002224, A002225, A002226, A002228.

More terms from Don Reble, Oct 12 2001

a(11)-a(12) from Sergey Paramonov, Apr 12 2024

Showing 1-10 of 19 results. Next

cp's OEIS Frontend

A002224 Smallest prime p of form p = 8k+1 such that first n primes (p_1=2, ..., p_n) are quadratic residues mod p.

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A000229 a(n) is the least number m such that the n-th prime is the least quadratic nonresidue modulo m.

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A002233 a(1) = 1; for n > 1, a(n) = least positive prime primitive root of n-th prime.

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A002225 a(n) is the smallest prime p such that each of the first n primes has three cube roots mod p.

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A002230 Primes with record values of the least positive primitive root.

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A002228 Smallest prime p such that first n primes (p_1=2, ..., p_n) are 11th power residues mod p.

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A147971 Indices of the records in the sequence of smallest positive quadratic nonresidues (A053760).

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A002199 Least negative primitive root of n-th prime.

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