cp's OEIS Frontend

A131139 Counts 2-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 A131139 #10 Jan 23 2023 15:08:25
%S A131139 1,1,4,5,36,40,145,180,1572,1712,6181,7712,43860,49856,171844,213953,
%T A131139 1634448,1798404,6362336,7945252,43391232,49532049,169120448,
%U A131139 210664996,1310330112,1471297572
%N A131139 Counts 2-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 A131139 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 A131139 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 A131139 The generating function is Product_{j>=0} theta_2(2^(2^j-1) x)^(2^j) where theta_2(y) is the generating function for 2-cores A010054  (this appears to be incorrect _Joerg Arndt_, Apr 06 2013)
%e A131139 a(2) = 4, since there are four quadratic algebras over Q_2 up to geometric equivalence, namely Q_2 times Q_2, Q_2(sqrt{-1}), Q_2(sqrt{2}) and Q_2(sqrt{-2})
%Y A131139 Cf. A000041, A010054, A131140.
%K A131139 nonn,more
%O A131139 0,3
%A A131139 David P. Roberts (roberts(AT)morris.umn.edu), Jun 19 2007