1,2

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.

Robin Visser, Table of n, a(n) for n = 1..3000

Jordi Guàrdia-Rúbies, John W. Jones, Kevin Keating, Sebastian Pauli, David P. Roberts, and David Roe, Families of p-adic fields, arXiv:2507.02360 [math.NT], 2025.

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=2, n=h*p^m, with gcd(h, p)=1, eps(-1)=-oo, eps(0)=0 and eps(s)=Sum_{i=1..s} 1/(p^i).

a(2)=7: 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, and one unramified extension with minimal polynomial x^2+x+1.

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^(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;

(SageMath)

def a(n):

ans = 0

for e in Integer(n).divisors():

s = Integer(e).valuation(2); ans += e

for i in range(1, s+1):

ans += e*2^(i+sum([n/(2^j) for j in range(1, i+1)]))

ans -= e*2^(i+sum([n/(2^j) for j in range(1, i)]))

return ans # Robin Visser, Nov 30 2025

Cf. A100977, A100978, A100979, A100980, A100981, A100983, A100984, A100985, A100986.

Sequence in context: A241359 A239602 A374295 * A152627 A277067 A354823

Adjacent sequences: A100973 A100974 A100975 * A100977 A100978 A100979

Volker Schmitt (clamsi(AT)gmx.net), Nov 24 2004

approved