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Search: id:A006167
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| A006167 |
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Number of factorization patterns of polynomials of degree n over F_2.
(Formerly M2349)
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+0
5
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| 1, 3, 4, 8, 11, 20, 27, 45, 61, 95, 128, 193, 257, 374, 497, 703, 927, 1287, 1683, 2297, 2987, 4013, 5186, 6887, 8843, 11614, 14836, 19294, 24514, 31622, 39968, 51167, 64377, 81839, 102509, 129528, 161539, 202959, 252124, 315110, 389949, 485062
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Let F_q(n) represent the number of factorization patterns of n with the property that there exists a monic polynomial V of degree n over the finite field F_q such that V factors over F_q into one of the F_q(n) factorization patterns. Sequence is for the q=2 case,
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REFERENCES
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R. A. Hultquist, G. L. Mullen and H. Niederreiter, Association schemes and derived PBIB designs of prime power order, Ars. Combin., 25 (1988), 65-82.
A. K. Agarwal and G. L. Mullen, Partitions with "d(a) copies of a", J. Combin. Theory, A48 (1988), 120-135.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
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FORMULA
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Euler transform of sequence b(n) = sum_{d|n, A001037(d)>=n/d} 1. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jun 19 2006
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EXAMPLE
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For n=3 there are 5 factorization patterns of cubic polynomials: 3, 2 + 1, 1^3, 1^2 + 1, 1 + 1 + 1. For example 1^2 + 1 corresponds to a cubic polynomial which factors as a linear of multiplicity 2 and a second distinct linear factor. For q=2 the pattern 1 + 1 + 1 is not allowed since over F_2 there are only two distinct monic irreducibles of degree 1. Thus a(3) = 4.
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CROSSREFS
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Cf. A006168-A006171.
Cf. A001037.
Sequence in context: A084421 A024786 A097497 this_sequence A137504 A109794 A034417
Adjacent sequences: A006164 A006165 A006166 this_sequence A006168 A006169 A006170
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KEYWORD
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nonn,nice
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AUTHOR
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njas
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EXTENSIONS
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Additional comments from Gary Mullen, Jun 03 2003.
More terms from Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jun 19 2006
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