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
Keywords
Links
- Jens Kruse Andersen, Table of n, a(n) for n = 1..10000
Programs
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Mathematica
With[{nn=60},Take[Union[Flatten[Table[p^Range[3,nn/3],{p,Prime[ Range[ nn]]}]]],nn]] (* Harvey P. Dale, Dec 10 2015 *) -
PARI
for(n=1, 10^6, if(isprimepower(n)>=3, print1(n, ", ")));
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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
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Python
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
Formula
Sum_{n>=1} 1/a(n) = Sum_{p prime} 1/(p^2*(p-1)) = A152441. - Amiram Eldar, Oct 24 2020
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