# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a100981 Showing 1-1 of 1 %I A100981 #9 Jan 09 2013 16:43:28 %S A100981 1,2,3,4,105,6,7,8,9,1210,11,12,13,14,9315,16,17,18,19,62420,21,22,23, %T A100981 24,8203025,26,27,28,29,2343630,31,32,33,34,13671735,36,37,38,39, %U A100981 78124840,41,42,43,44,439452945,46,47,48,49,295410156050,51 %N A100981 Number of totally ramified extensions over Q_5 with degree n in the algebraic closure of Q_5. %D A100981 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 A100981 a(n)=n*(sum_{s=0}^m p^s*(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 A100981 a(3)=3: there is one totally ramified extension with Galois group S_3, so there are 3 totally ramified extensions in the algebraic closure all isomorphic to Q_5[x]/(x^3+5) %p A100981 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^s*(p^(eps(p,s)*n)-p^(eps(p,s-1)*n)); od; a(n):=n*summe; %Y A100981 Cf. A100976, A100977, A100978, A100979, A100980, A100983, A100984, A100985, A100986. %K A100981 nonn %O A100981 1,2 %A A100981 Volker Schmitt (clamsi(AT)gmx.net), Nov 25 2004 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE