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%I #12 Jun 28 2020 02:54:34
%S 1,3,22,7,6,228,8,15,5323,18,12,5068,14,24,13092,31,18,1495839,20,42,
%T 157424,36,24,885660,31,42,942953404,56,30,9565848,32,63,19131816,54,
%U 48,24240086731,38,60,200884628,90,42,1033121184,44,84
%N Number of all extensions over Q_3 with degree n in the algebraic closure of Q_3.
%D M. Krasner, Le nombre des surcorps primitifs d'un degré donné et le nombre des surcorps métagaloisiens d'un degré donné d'un corps de nombres p-adiques. Comptes Rendus Hebdomadaires, Académie 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=3, 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+3, x^2-3 and one unramified x^2+2*x+2.
%p p:=3; 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, A100978, 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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