A191088 Least number k such that the first n primes have Kronecker symbol (p|k) = -1.

3, 5, 43, 43, 67, 67, 163, 163, 163, 163, 163, 163, 74093, 77683, 214037, 214037, 214037, 2004917, 36254563, 51599563, 96295483, 96295483, 114148483, 163520117, 163520117, 261153653, 261153653, 585811843

Offset

1,1

Comments

This implies, but is not equivalent to, that the first n primes are quadratic non-residues mod k. [Corrected by M. F. Hasler, Jan 18 2016]

The first 13 terms are prime. Sequence A191089 is similar, but forces k to be prime.

In case k is restricted to be a prime, the condition (p|k) = -1 becomes equivalent to saying that p is a quadratic non-residue (mod k). - M. F. Hasler, Jan 18 2016

Links

Table of n, a(n) for n=1..28.

Example

a(1) = 3 is the least number k such that prime(1) = 2 is not a square mod k (since for k=1 and k=2, p=2 would be zero, thus a square, mod k).
a(2) = 4 is the least number k such that prime(1) = 2 and prime(2) = 3 are not squares mod k (the only squares mod 4 are 0 = 0^1 = 2^2 and 1 = 1^2 = 3^3 (mod 4); while for k=2, p=2 would be zero, thus a square, and for k=3 this would be the case for p=3).

Mathematica

Table[k = 2; While[Length[Select[Prime[Range[n]], JacobiSymbol[#, k] == -1 &]] < n, k++]; k, {n, 15}]

Prog

(PARI) q=2; for(k=3, 1e9, forprime(p=2, q, if(kronecker(p, k)>=0, next(2))); print1(k", "); q=nextprime(q+1); k--) \\ Charles R Greathouse IV, Oct 10 2011

Crossrefs

Cf. A191089.

Sequence in context: A060433 A216085 A219018 * A191089 A199345 A062633

Adjacent sequences: A191085 A191086 A191087 * A191089 A191090 A191091

Keyword

nonn

Author

T. D. Noe, May 25 2011

Extensions

a(16)-a(28) from Charles R Greathouse IV, Oct 10 2011

Status

approved