A000229 a(n) is the least number m such that the n-th prime is the least quadratic nonresidue modulo m. (Formerly M2684 N1074)

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

1,1

Comments

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

References

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).

Links

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).

Example

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.

Mathematica

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 *)

Crossrefs

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

Differs from A002223, A045535 at 12th term.

Sequence in context: A066768 A225914 A062241 * A133435 A079061 A228724

Adjacent sequences: A000226 A000227 A000228 * A000230 A000231 A000232

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

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

Name edited by Thomas Ordowski, May 02 2019

Status

approved