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A088247 Orders of proper semifields.

%I A088247 #34 Sep 11 2024 23:47:27
%S A088247 16,27,32,64,81,125,128,243,256,343,512,625,729,1024,1331,2048,2187,
%T A088247 2197,2401,3125,4096,4913,6561,6859,8192,12167,14641,15625,16384,
%U A088247 16807,19683,24389,28561,29791,32768,50653,59049,65536,68921,78125,79507
%N A088247 Orders of proper semifields.
%C A088247 Subset of prime powers A000961. Superset of orders of twisted fields A088248.
%C A088247 Prime powers p^e > 8 with e > 2, thus excluding the primes, the semiprimes, unity and 8. _Robert G. Wilson v_, Mar 11 2014
%D A088247 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.
%H A088247 Chai Wah Wu, <a href="/A088247/b088247.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)
%H A088247 Hauke Klein, <a href="http://www.math.uni-kiel.de/geometrie/klein/math/geometry/semi.html">Semifields</a>, provides definition, context, links, theorem.
%H A088247 D. E. Knuth, <a href="http://resolver.caltech.edu/CaltechETD:etd-06042004-141331">Finite semifields and projective planes</a>, Caltech PhD dissertation, library online PDF version.
%F A088247 All p^k >= 16, prime p, k >= 3.
%F A088247 a(n) = n^3 log^3 n + O(n^3 log^2 n log log n). - _Charles R Greathouse IV_, Mar 11 2014
%t A088247 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 *)
%t A088247 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 *)
%o A088247 (PARI) is(n)=isprimepower(n)>2 && n>8 \\ _Charles R Greathouse IV_, Mar 11 2014
%o A088247 (Python)
%o A088247 from math import isqrt
%o A088247 from sympy import primerange, integer_nthroot, primepi
%o A088247 def A088247(n):
%o A088247     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 A088247     def f(x): return int(n+1+x-sum(primepi(integer_nthroot(x, k)[0]) for k in range(3, x.bit_length())))
%o A088247     def bisection(f,kmin=0,kmax=1):
%o A088247         while f(kmax) > kmax: kmax <<= 1
%o A088247         while kmax-kmin > 1:
%o A088247             kmid = kmax+kmin>>1
%o A088247             if f(kmid) <= kmid:
%o A088247                 kmax = kmid
%o A088247             else:
%o A088247                 kmin = kmid
%o A088247         return kmax
%o A088247     return bisection(f,n,n) # _Chai Wah Wu_, Sep 11 2024
%Y A088247 Cf. A000961, A088248.
%K A088247 nonn,easy,nice
%O A088247 1,1
%A A088247 _Marc LeBrun_, Sep 25 2003