A131139 -id:A131139 - 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.

1, 1, 2, 9, 11, 19, 83, 99, 172, 1100, 1244, 2250, 8687, 10683, 18173, 67950, 82785, 140825, 665955, 780030, 1367543, 4867750, 6027860, 10149291, 35453711, 43581422

Offset: 0

David P. Roberts (roberts(AT)morris.umn.edu), Jun 19 2007

In general, the number of p-wild partitions of n is equal to the number of partitions of n if and only if n

			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.

David P. Roberts, Wild Partitions and Number Theory. Journal of Integer Sequences, Volume 10, Issue 6, Article 07.6.6, (2007).

Cf. A000041, A033687, A131139.

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]

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