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%I #6 Jan 09 2013 16:41:24
%S 1,3,4,7,106,12,8,15,13,1818,12,28,14,24,12424,31,18,39,20,109242,32,
%T 36,24,60,8281131,42,40,56,30,4687272,32,63,48,54,15624848,91,38,60,
%U 56,146484090,42,96,44,84,634765378,72,48,124,57
%N Number of all extensions over Q_5 with degree n in the algebraic closure of Q_5.
%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)=(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=5, 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)=3 There are 2 ramified extensions with minimal polynomials x^2-5, x^2-10 and one unramified x^2+4*x+2.
%p p:=5; 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 Cf. A100976, A100977, A100979, A100980, A100981, A100983, A100984, A100985, A100986.
%K nonn
%O 1,2
%A Volker Schmitt (clamsi(AT)gmx.net), Nov 24 2004
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