A088248 Orders of twisted fields.

27, 64, 81, 125, 243, 256, 343, 512, 625, 729, 1024, 1331, 2187, 2197, 2401, 3125, 4096, 4913, 6561, 6859, 12167, 14641, 15625, 16384, 16807, 19683, 24389, 28561, 29791, 32768, 50653, 59049, 65536, 68921, 78125, 79507, 83521, 103823, 117649

Offset: 1

Author

Marc LeBrun, Sep 25 2003

Keywords

• easy
• nonn
• nice

Comments

Subset of prime powers A000961. Subset of orders of semifields A088247.

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, p336.

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
• D. E. Knuth, Finite Semifields and Projective Planes, Journal of Algebra, Volume 2, Issue 2, June 1965, Pages 182-217.

Crossrefs

Cf. A000961, A088247.

Programs

Mathematica

okQ[n_] := Module[{f, p, k}, If[n <= 16, False, f = FactorInteger[n]; If[Length[f] > 1, False, {p, k} = First[f]; k >= 3 && Not[p == 2 && PrimeQ[k]]]]]; Select[Range[10^6], okQ] (* Jean-François Alcover, Jul 07 2015 *)

Python

from math import isqrt
from sympy import primerange, integer_nthroot, primepi
def A088248(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n+x+primepi(x.bit_length()-1)-sum(primepi(integer_nthroot(x, k)[0]) for k in range(3, x.bit_length())))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f,n,n) # Chai Wah Wu, Sep 11 2024

Formula

All p^k > 16, prime p, k>=3, except 2^q, q prime.