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%I #6 Jan 09 2013 16:41:38
%S 1,6,3,92,5,90,7,5880,9,630,11,23028,13,3570,15,6021104,17,18414,19,
%T 2580460,21,90090,23,377290728,25,425958,27,233963492,29,1966050,31,
%U 1578396286944,33,8912862,35,19308478428,37,39845850,39,108
%N Number of totally ramified extensions over Q_2 with degree n in the algebraic closure of Q_2.
%D M. Krasner, Le nombre des surcorps primitifs d'un degre donne et le nombre des surcorps metagaloisiens d'un degre donne d'un corps de nombres p-adiques. Comptes Rendus Hebdomadaires, Academie des Sciences, Paris 254, 255, 1962
%F a(n)=n*(sum_{s=0}^m p^s*(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 a(2)=6 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, there is another one by x^2+x+1, but this is unramified.
%p 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^s*(p^(eps(p,s)*n)-p^(eps(p,s-1)*n)); od; a(n):=n*summe;
%Y Cf. A100976, A100977, A100978, A100980, A100981, A100983, A100984, A100985, A100986.
%K nonn
%O 1,2
%A Volker Schmitt (clamsi(AT)gmx.net), Nov 25 2004
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