OFFSET
1,1
COMMENTS
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
Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
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
(Python)
from math import isqrt
from sympy import primerange, integer_nthroot, primepi
def A088247(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+1+x-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
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
Marc LeBrun, Sep 25 2003
STATUS
approved