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Search: id:A131140
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| A131140 |
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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. |
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| 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
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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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.
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LINKS
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David P. Roberts, Wild Partitions and Number Theory Journal of Integer Sequences, Volume 10, Issue 6, Article 07.6.6, (2007)
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FORMULA
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The generating function is prod_{j=0}^infinity theta_3(2^((3^j-1)/2) x)^(3^j) where theta_3(y) is the generating function for 3-cores A033687
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EXAMPLE
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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.
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CROSSREFS
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Cf. A000041, A033687, A131139.
Sequence in context: A098934 A043307 A049343 this_sequence A022114 A041099 A041581
Adjacent sequences: A131137 A131138 A131139 this_sequence A131141 A131142 A131143
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KEYWORD
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nonn
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AUTHOR
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David P. Roberts (roberts(AT)morris.umn.edu), Jun 19 2007
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