1,2

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,

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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.

T. D. Noe, Table of n, a(n) for n=1..1000

Euler transform of sequence b(n) = sum_{d|n, A001037(d)>=n/d} 1. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jun 19 2006

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.

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

nonn,nice

N. J. A. Sloane (njas(AT)research.att.com).

Additional comments from Gary Mullen, Jun 03 2003.

More terms from Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jun 19 2006