%I #3 May 04 2024 11:46:37

%S 2,7,11,2,11,5,3,7,2,3,5,11,3,5,7,2,2,7,3,11,5,3,7,5,2,5,11,3,5,7,3,7,

%T 2,3,13,2,3,11,5,3,2,11,3,5,11,3,11,11,2,7,5,3,7,5,3,5,2,3,5,7,3,13,3,

%U 2,2,5,3,2,5,3,5,7,2,5,11

%N a(n) = smallest prime p such that x^2-n has roots in the p-adic integers.

%e a(6)=5 because x^2-6 has roots in the 5-adic integers. Roots are

%e 4+5+4*5^2+2*5^4+3*5^5+2*5^6+5^7+3*5^8+O(5^9) and

%e 1+3*5+4*5^3+2*5^4+5^5+2*5^6+3*5^7+5^8+O(5^9); but this is irreducible over Qp for p in {2,3} (x^2-6 is Eisenstein for p=2 and 3).

%p A100020 := proc(n)

%p local p,anz ;

%p p := 1 ;

%p anz := 0 ;

%p while anz =0 do

%p p := nextprime(p) ;

%p poly := x^2-n ;

%p anz := nops([padic[rootp](poly, p)]);

%p end do:

%p p ;

%p end proc:

%p seq(A100020(n),n=1..100) ;

%Y Cf. A099408.

%K nonn

%O 1,1

%A Volker Schmitt (clamsi(AT)gmx.net), Nov 19 2004