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A100979 Number of totally ramified extensions over Q_2 with degree n in the algebraic closure of Q_2. 8
1, 6, 3, 92, 5, 90, 7, 5880, 9, 630, 11, 23028, 13, 3570, 15, 6021104, 17, 18414, 19, 2580460, 21, 90090, 23, 377290728, 25, 425958, 27, 233963492, 29, 1966050, 31, 1578396286944, 33, 8912862, 35, 19308478428, 37, 39845850, 39, 108 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

REFERENCES

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

LINKS

Table of n, a(n) for n=1..40.

FORMULA

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)

EXAMPLE

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.

MAPLE

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;

CROSSREFS

Cf. A100976, A100977, A100978, A100980, A100981, A100983, A100984, A100985, A100986.

Sequence in context: A324673 A324004 A002610 * A279107 A266661 A136133

Adjacent sequences:  A100976 A100977 A100978 * A100980 A100981 A100982

KEYWORD

nonn

AUTHOR

Volker Schmitt (clamsi(AT)gmx.net), Nov 25 2004

STATUS

approved

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Last modified October 21 11:35 EDT 2019. Contains 328296 sequences. (Running on oeis4.)