A190898 Least odd prime p>n^2 with (n/p) = 1, where ( / ) is the Legendre symbol

3, 7, 11, 17, 29, 43, 53, 71, 83, 107, 127, 157, 173, 199, 229, 257, 293, 337, 379, 401, 457, 499, 541, 577, 631, 683, 733, 787, 857, 911, 967, 1031, 1091, 1163, 1229, 1297, 1373, 1447, 1553, 1601, 1697, 1787, 1867, 1973, 2029, 2129, 2213, 2339, 2411, 2503, 2617, 2707, 2819, 2927, 3041, 3137, 3251, 3457, 3491, 3607

Offset

1,1

Comments

Conjecture: a(n)<(n+1)^2 for all n>0. (See also A185150.)

This conjecture implies that a(1),a(2),a(3),... are pairwise distinct.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Example

a(2)=7 since 7 is the first prime p>2^2 with (2/p) = 1.

Mathematica

Do[Do[If[n^2+k>2&&PrimeQ[n^2+k]==True&&JacobiSymbol[n, n^2+k]==1, Print[n, " ", n^2+k]; Goto[aa]], {k, 1, 2n}];
Label[aa]; Continue, {n, 1, 100}]
js[n_]:=Module[{p=NextPrime[n^2]}, While[JacobiSymbol[n, p]!=1, p= NextPrime[ p]]; p]; Join[{3}, Array[js, 60, 2]] (* Harvey P. Dale, Jan 29 2023 *)

Crossrefs

Cf. A180150, A014085, A185636.

Sequence in context: A038949 A106967 A045420 * A142248 A045421 A072456

Adjacent sequences: A190895 A190896 A190897 * A190899 A190900 A190901

Keyword

nonn,nice

Author

Zhi-Wei Sun, Dec 29 2012

Status

approved