cp's OEIS Frontend

A131140 Counts 3-wild partitions. In general p-wild partitions of n are defined so that they are in bijection with geometric equivalence classes of degree n algebra extensions of the p-adic field Q_p. Here two algebra extensions are equivalent if they become isomorphic after tensoring with the maximal unramified extension of Q_p.

%I A131140 #14 Jan 28 2023 12:18:47
%S A131140 1,1,2,9,11,19,83,99,172,1100,1244,2250,8687,10683,18173,67950,82785,
%T A131140 140825,665955,780030,1367543,4867750,6027860,10149291,35453711,
%U A131140 43581422
%N A131140 Counts 3-wild partitions. In general p-wild partitions of n are defined so that they are in bijection with geometric equivalence classes of degree n algebra extensions of the p-adic field Q_p. Here two algebra extensions are equivalent if they become isomorphic after tensoring with the maximal unramified extension of Q_p.
%C A131140 In general, the number of p-wild partitions of n is equal to the number of partitions of n if and only if n<p. From n=p onward, there are many more p-wild partitions.
%H A131140 David P. Roberts, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Roberts/wildpart2.html">Wild Partitions and Number Theory</a>. Journal of Integer Sequences, Volume 10, Issue 6, Article 07.6.6, (2007).
%F A131140 The generating function is Product_{j>=0} theta_3(2^((3^j-1)/2)*x)^(3^j) where theta_3(y) is the generating function for 3-cores A033687. [This appears to be incorrect - _Joerg Arndt_, Apr 06 2013]
%e A131140 a(3) = 9, since there are four quadratic algebras over Q_3 up to geometric equivalence, namely the unramified algebra Q_3 times Q_3 times Q_3, the tamely ramified algebras Q_3 times Q_3[x]/(x^2-3) and two, two and three wildly ramified algebras with discriminants 3^3, 3^4 and 3^5 respectively.
%Y A131140 Cf. A000041, A033687, A131139.
%K A131140 nonn,more
%O A131140 0,3
%A A131140 David P. Roberts (roberts(AT)morris.umn.edu), Jun 19 2007