A088247 -id:A088247 - cp's OEIS frontend

A246549 Prime powers p^e where p is a prime and e >= 3 (prime powers without 1, the primes, or the squares of primes).

8, 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, 83521, 103823, 117649, 130321, 131072, 148877, 161051, 177147, 205379

Offset: 1

Joerg Arndt, Aug 29 2014

nonn

Consists of 8 and the terms of A088247. - R. J. Mathar, Sep 01 2014

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A000961, A152441, A246547, A246550.

With[{nn=60},Take[Union[Flatten[Table[p^Range[3,nn/3],{p,Prime[ Range[ nn]]}]]],nn]] (* Harvey P. Dale, Dec 10 2015 *)

for(n=1, 10^6, if(isprimepower(n)>=3, print1(n, ", ")));

m=10^6; v=[]; forprime(p=2, m^(1/3), e=3; while(p^e<=m, v=concat(v, p^e); e++)); v=vecsort(v) \\ Faster program. Jens Kruse Andersen, Aug 29 2014

from math import isqrt
from sympy import primerange, integer_nthroot, primepi
def A246549(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-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

Sum_{n>=1} 1/a(n) = Sum_{p prime} 1/(p^2*(p-1)) = A152441. - Amiram Eldar, Oct 24 2020

A088248 Orders of twisted fields.

Original entry on oeis.org

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

Marc LeBrun, Sep 25 2003

Subset of prime powers A000961. Subset of orders of semifields 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, p336.

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.

Cf. A000961, A088247.

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 *)

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

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

cp's OEIS Frontend

A246549 Prime powers p^e where p is a prime and e >= 3 (prime powers without 1, the primes, or the squares of primes).

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