A002224 -id:A002224 - cp's OEIS frontend

A002223 Smallest prime p of form p = 8k-1 such that first n primes (p_1=2, ..., p_n) are quadratic residues mod p.

7, 23, 71, 311, 479, 1559, 5711, 10559, 18191, 31391, 366791, 366791, 366791, 4080359, 12537719, 30706079, 36415991, 82636319, 120293879, 120293879, 131486759, 131486759, 2929911599, 2929911599, 7979490791, 33857579279

Offset: 1

N. J. A. Sloane

			12^2 = 2 mod 71, 28^2 = 3 mod 71, 17^2 = 5 mod 71.

N. D. Bronson and D. A. Buell, Congruential sieves on FPGA computers, pp. 547-551 of Mathematics of Computation 1943-1993 (Vancouver, 1993), Proc. Symp. Appl. Math., Vol. 48, Amer. Math. Soc. 1994.
D. H. Lehmer, E. Lehmer and D. Shanks, Integer sequences having prescribed quadratic character, Math. Comp., 24 (1970), 433-451.
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.

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. A045535, A002224, A002225.

np[] := While[p = NextPrime[p]; Mod[p, 8] != 7]; p = 2; A002223 = {}; pp = {2}; np[]; While[ Length[A002223] < 26, If[Union[ JacobiSymbol[#, p] &[pp]] === {1}, AppendTo[pp, NextPrime[Last[pp]]]; Print[p]; AppendTo[A002223, p], np[]]]; A002223 (* Jean-François Alcover, Sep 09 2011 *)

The Bronson-Buell reference gives terms through 227.

More terms from Don Reble, Sep 19 2001

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

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

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

A094928 Let p = n-th prime == 1 mod 8 (A007519); a(n) = smallest prime q such that p is not a square mod q.

Original entry on oeis.org

3, 3, 5, 3, 5, 3, 3, 5, 3, 7, 3, 3, 5, 5, 3, 3, 7, 5, 3, 5, 3, 3, 5, 3, 7, 3, 3, 5, 3, 7, 3, 3, 3, 3, 5, 3, 3, 11, 5, 3, 3, 11, 5, 3, 11, 3, 7, 3, 5, 7, 3, 3, 3, 3, 7, 3, 3, 7, 5, 3, 3, 5, 5, 11, 5, 3, 3, 5, 5, 3, 7, 5, 3, 5, 3, 7, 3, 7, 3, 5, 3, 3, 3, 5, 11, 5, 3, 5, 3, 3, 13, 5, 3, 3, 3, 3, 5, 5, 3, 5, 3, 7

Offset: 1

N. J. A. Sloane, Jun 19 2004

			n=3, p = 73, a(3) = q = 5: Legendre(73,5) = -1.

M. Kneser, Quadratische Formen, Springer, 2002; see Hilfssatz 18.3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007519, A002224, A144294, A269704.

Subsequence of A094929.

f:= proc(p) local q;
     q:= 3:
     do
      if numtheory:-quadres(p,q) = -1 then return q fi;
      q:= nextprime(q);
     od;
end proc:
map(f, select(isprime, [seq(p,p=1..10000,8)])); # Robert Israel, May 06 2019

f[n_] := Prime[ Position[ JacobiSymbol[n, Select[Range[3, n - 1], PrimeQ[ # ] &]], -1][[1, 1]] + 1]; f /@ Select[ Prime[ Range[435]], Mod[ #, 8] == 1 &] (* Robert G. Wilson v, Jun 23 2004 *)

a(n) = A094929(A269704(n)). - Robert Israel, May 06 2019

More terms from Robert G. Wilson v, Jun 23 2004

A147969 Smallest prime p modulo which numbers 1,2,...,n are quadratic residues.

Original entry on oeis.org

2, 7, 23, 23, 71, 71, 311, 311, 311, 311, 479, 479, 1559, 1559, 1559, 1559, 5711, 5711, 10559, 10559, 10559, 10559, 18191, 18191, 18191, 18191, 18191, 18191, 31391, 31391, 366791, 366791, 366791, 366791, 366791, 366791, 366791, 366791, 366791

Offset: 1

Max Alekseyev, Nov 18 2008

nonn

The same primes without repetitions are listed in A147970.

Charles R Greathouse IV, Table of n, a(n) for n = 1..100

Cf. A000229, A002223, A002224, A045535, A053760, A133435, A147970, A147971.

a(n)=forprime(p=2,default(primelimit),forprime(i=2,n, if(kronecker(i,p)<1,next(2)));return(p)) \\ Charles R Greathouse IV, Apr 06 2012

A205532 Primes at which occur records of A205531 and A205535.

Original entry on oeis.org

2, 13, 61, 109, 1009, 2689, 8089, 33049, 53881, 87481, 483289, 515761, 1083289, 3818929, 9257329

Offset: 1

M. F. Hasler, Jan 28 2012

nonn

Related to the 4.X Selfridge Conjecture by P. Underwood, cf. link.
Records occur at [A205532(n), A205534(n)] = [2, 1], [13, 3], [61, 5], [109, 6], [1009, 9], [2689, 11], [8089, 15], [33049, 17], [53881, 21], [87481, 27], [483289, 29], [515761, 35], [1083289, 39], [3818929, 45], [9257329, 51],...
It appears that a(n)=A102295(n+1) for n>4; they are also terms of A002224 and A096637.

P. Underwood, 4.X Selfridge Conjecture (on "Prime Pages" profile), Jan 2012.

m=-1; forprime(p=1,default(primelimit), if(m+0A205535(p),m), print1(p",")))

A147970 Primes corresponding to the records in the sequence of smallest positive quadratic nonresidues (A053760).

Original entry on oeis.org

2, 7, 23, 71, 311, 479, 1559, 5711, 10559, 18191, 31391, 366791, 3818929, 9257329, 22000801, 36415991, 48473881, 120293879, 131486759, 2929911599, 7979490791, 23616331489, 89206899239, 121560956039, 196265095009, 513928659191, 5528920734431, 8402847753431, 70864718555231

Offset: 1

Max Alekseyev, Nov 18 2008

nonn

Cf. A000229, A002223, A002224, A045535, A053760, A133435, A147969, A147971.

Prime p=A000040(n) is in this sequence iff A053760(k) < A053760(n) for every kA000040(A147971(n))

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

A096637 Smallest prime p == 1 mod 8 (A007519) and p > prime(n+2) such that p is a quadratic residue mod the first n odd primes 3, 5, 7, 11, ..., prime(n+1), and p is a quadratic non-residue mod prime(n+2).

Original entry on oeis.org

17, 73, 241, 1009, 2689, 8089, 33049, 53881, 87481, 483289, 515761, 1083289, 7921489, 3818929, 9257329, 22000801, 68204761, 48473881, 175244281, 1149374521, 427733329, 898716289

Offset: 0

Robert G. Wilson v, Jun 24 2004

nonn

Same as smallest prime p == 1 mod 8 with the property that the Legendre symbol (p|q) = 1 for the first n odd primes q = prime(k+1), k = 1, 2, ..., n, and (p|q) = -1 for q = prime(n+2).

Cf. A094928, A002224, A096636.

f[n_] := Block[{k = 2}, While[ JacobiSymbol[n, Prime[k]] == 1, k++ ]; Prime[k]]; t = Table[0, {50}]; Do[p = Prime[n]; If[Mod[p, 8] == 1, a = f[p]; If[ t[[ PrimePi[a]]] == 0, t[[ PrimePi[a]]] = p; Print[ PrimePi[a], " = ", p]]], {n, 10^9}]; t

Better name from Jonathan Sondow, Mar 07 2013

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cp's OEIS Frontend

A002223 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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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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A147971 Indices of the records in the sequence of smallest positive quadratic nonresidues (A053760).

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

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

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A094928 Let p = n-th prime == 1 mod 8 (A007519); a(n) = smallest prime q such that p is not a square mod q.

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A147969 Smallest prime p modulo which numbers 1,2,...,n are quadratic residues.

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PARI

A205532 Primes at which occur records of A205531 and A205535.

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PARI