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

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Examples

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

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).
• 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.

Links

• 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]

Crossrefs

Cf. A002223, A002225, A002226.

Programs

Mathematica

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 *)

PARI

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

Extensions

More terms from Don Reble, Sep 19 2001

More terms from Mike Oakes, Nov 28 2022