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

Author

Zhi-Wei Sun, Dec 29 2012

Keywords

• nonn
• nice

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.

Examples

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

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.

Crossrefs

Cf. A180150, A014085, A185636.

Programs

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 *)