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

%I A246549 #26 Sep 11 2024 22:48:52
%S A246549 8,16,27,32,64,81,125,128,243,256,343,512,625,729,1024,1331,2048,2187,
%T A246549 2197,2401,3125,4096,4913,6561,6859,8192,12167,14641,15625,16384,
%U A246549 16807,19683,24389,28561,29791,32768,50653,59049,65536,68921,78125,79507,83521,103823,117649,130321,131072,148877,161051,177147,205379
%N A246549 Prime powers p^e where p is a prime and e >= 3 (prime powers without 1, the primes, or the squares of primes).
%C A246549 Consists of 8 and the terms of A088247. - _R. J. Mathar_, Sep 01 2014
%H A246549 Jens Kruse Andersen, <a href="/A246549/b246549.txt">Table of n, a(n) for n = 1..10000</a>
%F A246549 Sum_{n>=1} 1/a(n) = Sum_{p prime} 1/(p^2*(p-1)) = A152441. - _Amiram Eldar_, Oct 24 2020
%t A246549 With[{nn=60},Take[Union[Flatten[Table[p^Range[3,nn/3],{p,Prime[ Range[ nn]]}]]],nn]] (* _Harvey P. Dale_, Dec 10 2015 *)
%o A246549 (PARI) for(n=1, 10^6, if(isprimepower(n)>=3, print1(n, ", ")));
%o A246549 (PARI) 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
%o A246549 (Python)
%o A246549 from math import isqrt
%o A246549 from sympy import primerange, integer_nthroot, primepi
%o A246549 def A246549(n):
%o A246549     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 A246549     def f(x): return int(n+x-sum(primepi(integer_nthroot(x, k)[0]) for k in range(3, x.bit_length())))
%o A246549     def bisection(f,kmin=0,kmax=1):
%o A246549         while f(kmax) > kmax: kmax <<= 1
%o A246549         while kmax-kmin > 1:
%o A246549             kmid = kmax+kmin>>1
%o A246549             if f(kmid) <= kmid:
%o A246549                 kmax = kmid
%o A246549             else:
%o A246549                 kmin = kmid
%o A246549         return kmax
%o A246549     return bisection(f,n,n) # _Chai Wah Wu_, Sep 11 2024
%Y A246549 Cf. A000961, A152441, A246547, A246550.
%K A246549 nonn
%O A246549 1,1
%A A246549 _Joerg Arndt_, Aug 29 2014