A133435 -id:A133435 - cp's OEIS frontend

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

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

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

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

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

A147972 Smallest prime p modulo which the first n primes are nonzero quadratic residues.

Original entry on oeis.org

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

Offset: 1

Max Alekseyev, Nov 18 2008

nonn

The same primes without repetitions are listed in A147970.
a(n) <= min{A002223(n), A002224(n)}. What is the smallest n for which this inequality is strict?
By definition, a(n) == 1, 7 (mod 8), so a(n) = min{A002223(n), A002224(n)}. - Jianing Song, Feb 18 2019

Cf. A000229, A002189, A002223, A002224, A045535, A053760, A133435, A147969, A147970, A147971.

Smallest prime p such that each of the first n primes has q q-th roots mod p: this sequence (q=2), A002225 (q=3), A002226 (q=5), A002227 (q=7), A002228 (q=11), A060363 (q=13), A060364 (q=17).

(*version 7.0*)m=1;P=7;Lst={p};While[m<25,m++;S=Prime[Range[m]];While[MemberQ[JacobiSymbol[S,p],-1],p=NextPrime[p]];Lst=Append[Lst,P]];Lst (* Emmanuel Vantieghem, Jan 31 2012 *)

t=2;forprime(p=2,1e9,forprime(q=2,t,if(kronecker(q,p)<1,next(2)));print1(p", ");t=nextprime(t+1);p--) \\ Charles R Greathouse IV, Jan 31 2012

a(n) >= min{A002189(n-1), A045535(n-1)}. - Jianing Song, Feb 18 2019

a(23)-a(25) from Emmanuel Vantieghem, Jan 31 2012

a(26)-a(37) from Max Alekseyev, Aug 21 2015

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

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

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

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Author

Keywords

Crossrefs

Formula

Extensions

A147972 Smallest prime p modulo which the first n primes are nonzero quadratic residues.

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Author

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Programs

Mathematica

PARI

Formula

Extensions