Search: id:A100976
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%I A100976
%S A100976 1,7,4,107,6,124,8,6835,13,762,12,31724,14,4088,24,6999011,18,26611,20,
%T A100976 3121122,32,98292,24,519765964,31,458738,40,267911128,30,3145704,32,
%U A100976 1834748739523,48,9437166,48,27903655871,38,41943020,56
%N A100976 Number of all extensions over Q_2 with degree n in the algebraic closure 
               of Q_2.
%D A100976 M. Krasner, Le nombre des surcorps primitifs d'un degre donne et le nombre 
               des surcorps metagaloisiens d'un degre donne d'un corp de nombre 
               p-adique. Comptes Redus Hebdomadaires, Academie des Science, Paris 
               254, 255, 1962
%F A100976 a(n)=(sum_{d|h}d)*(sum_{s=0}^m (p^(m+s+1)-p^(2*s))/(p-1)*(p^(eps(s)*n)-p^(eps(s-1)*n))), 
               where p=2, n=h*p^m, with gcd(h, p)=1, eps(-1)=-infinity, eps(0)=0 
               and eps(s)=sum_{i=1 to s} 1/(p^i)
%e A100976 a(2)=7: There are 6 ramified extensions with minimal polynomials x^2+2, 
               x^2-2, x^2+6, x^2-6, x^2+2x+2, x^2+2x+6 and one unramified x^2+x+1.
%p A100976 p:=2; eps:=proc()local p,s,i,sum; p:=args[1]; s:=args[2]; if s=-1 then 
               return -infinity; fi; if s=0 then return 0; fi; sum:=0; for i from 
               1 to s do sum:=sum+1/p^i; od; return sum; end: ppart:=proc() local 
               p,n; p:=args[1]; n:=args[2]; return igcd(n,p^n); end: qpart:=proc() 
               local p,n; p:=args[1]; n:=args[2]; return n/igcd(n,p^n); end: logp:=proc() 
               local p, pp; p:=args[1]; pp:=args[2]; if op(ifactors(pp))[2]=[] then 
               return 0; else return op(op(ifactors(pp))[2])[2]; fi; end: summe:=0; 
               m:=logp(p, ppart(p,n)); h:=qpart(p,n); for s from 0 to m do summe:=summe+(p^(m+s+1)-p^(2*s))/
               (p-1)*(p^(eps(p,s)*n)-p^(eps(p,s-1)*n)); od; a(n):=sigma(h)*summe;
%Y A100976 Cf. A100977, A100978, A100979, A100980, A100981, A100983, A100984, A100985, 
               A100986.
%Y A100976 Sequence in context: A070427 A140721 A038270 this_sequence A152627 A113223 
               A096414
%Y A100976 Adjacent sequences: A100973 A100974 A100975 this_sequence A100977 A100978 
               A100979
%K A100976 nonn
%O A100976 1,2
%A A100976 Volker Schmitt (clamsi(AT)gmx.net), Nov 24 2004

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