A088247 Orders of proper semifields.

16, 27, 32, 64, 81, 125, 128, 243, 256, 343, 512, 625, 729, 1024, 1331, 2048, 2187, 2197, 2401, 3125, 4096, 4913, 6561, 6859, 8192, 12167, 14641, 15625, 16384, 16807, 19683, 24389, 28561, 29791, 32768, 50653, 59049, 65536, 68921, 78125, 79507

Offset

1,1

Comments

Subset of prime powers A000961. Superset of orders of twisted fields A088248.

The Knuth reference is from his PhD dissertation under Marshall Hall, Jr., at Caltech, Jan 01, 1963. Knuth's "paper makes contributions to the structure theory of finite semifields, i.e., of finite nonassociative division algebras with unit. It is shown that a semifield may be conveniently represented as a 3-dimensional array of numbers and that matrix multiplications applied to each of the three dimensions correspond to the concept of isotopy. The six permutations of three coordinates yield a new way to obtain projective planes from a given plane. Several new classes of semifields are constructed; in particular one class, called the binary semifields, provides an affirmative answer to the conjecture that there exist non-Desarguesian projective planes of all orders 2[...], if n is greater than 3. With the advent of binary semifields, the gap between necessary and sufficient conditions on the possible orders of semifields has disappeared." - Jonathan Vos Post, Dec 30 2007

Prime powers p^e > 8 with e > 2, thus excluding the primes, the semiprimes, unity and 8. Robert G. Wilson v, Mar 11 2014

References

D. E. Knuth, "Finite Semifields and Projective Planes", Selected Papers on Discrete Mathematics, Center for the Study of Language and Information, Leland Stanford Junior University, CA, 2003, p. 335.

Links

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

Hauke Klein, Semifields, provides definition, context, links, theorem.

D. E. Knuth, Finite semifields and projective planes, Caltech PhD dissertation, library online PDF version.

Formula

All p^k >= 16, prime p, k >= 3.

a(n) = n^3 log^3 n + O(n^3 log^2 n log log n). - Charles R Greathouse IV, Mar 11 2014

Mathematica

max = 10^5; Clear[f]; f[2] = {}; p = Prime /@ Range[PrimePi[max^(1/3) // N]]; f[k_] := f[k] = Select[Union[f[k-1], p^k], # < max &]; f[k = 3]; While[f[k] != f[k-1], k++]; f[k] // Rest (* Jean-François Alcover, Sep 26 2013 *)
Select[ Range[ 9, 80000 ], PrimeOmega@# > 2 && Mod[ #, # - EulerPhi@# ] == 0 & ] (* or *) mx = 80000; Rest@ Sort@ Flatten@ Table[ Prime[n]^e, {n, PrimePi[ mx^(1/3)]}, {e, 3, Log[ Prime@ n, mx]}] (* Robert G. Wilson v, Mar 11 2014 *)

Prog

(PARI) is(n)=isprimepower(n)>2 && n>8 \\ Charles R Greathouse IV, Mar 11 2014

Crossrefs

Cf. A000961, A088248.

Sequence in context: A152444 A199013 A118642 * A366962 A032610 A286429

Adjacent sequences: A088244 A088245 A088246 * A088248 A088249 A088250

Keyword

nonn,easy,nice

Author

Marc LeBrun, Sep 25 2003

Status

approved