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

Author

N. J. A. Sloane

Keywords

• nonn
• nice

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

Examples

			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.
		

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).

Crossrefs

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

Differs from A002223, A045535 at 12th term.

Programs

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 *)

Extensions

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

Name edited by Thomas Ordowski, May 02 2019