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%I #18 Sep 11 2024 23:47:30
%S 27,64,81,125,243,256,343,512,625,729,1024,1331,2187,2197,2401,3125,
%T 4096,4913,6561,6859,12167,14641,15625,16384,16807,19683,24389,28561,
%U 29791,32768,50653,59049,65536,68921,78125,79507,83521,103823,117649
%N Orders of twisted fields.
%C Subset of prime powers A000961. Subset of orders of semifields A088247.
%D 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.
%H Chai Wah Wu, <a href="/A088248/b088248.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)
%H D. E. Knuth, <a href="http://dx.doi.org/10.1016/0021-8693(65)90018-9">Finite Semifields and Projective Planes</a>, Journal of Algebra, Volume 2, Issue 2, June 1965, Pages 182-217.
%F All p^k > 16, prime p, k>=3, except 2^q, q prime.
%t 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 *)
%o (Python)
%o from math import isqrt
%o from sympy import primerange, integer_nthroot, primepi
%o def A088248(n):
%o 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)))
%o 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())))
%o def bisection(f,kmin=0,kmax=1):
%o while f(kmax) > kmax: kmax <<= 1
%o while kmax-kmin > 1:
%o kmid = kmax+kmin>>1
%o if f(kmid) <= kmid:
%o kmax = kmid
%o else:
%o kmin = kmid
%o return kmax
%o return bisection(f,n,n) # _Chai Wah Wu_, Sep 11 2024
%Y Cf. A000961, A088247.
%K easy,nonn,nice
%O 1,1
%A _Marc LeBrun_, Sep 25 2003