A246550 Prime powers p^e where p is a prime and e >= 4.

16, 32, 64, 81, 128, 243, 256, 512, 625, 729, 1024, 2048, 2187, 2401, 3125, 4096, 6561, 8192, 14641, 15625, 16384, 16807, 19683, 28561, 32768, 59049, 65536, 78125, 83521, 117649, 130321, 131072, 161051, 177147, 262144, 279841, 371293, 390625, 524288, 531441, 707281, 823543, 923521, 1048576, 1419857, 1594323, 1771561

Offset: 1

Author

Joerg Arndt, Aug 29 2014

Keywords

• nonn

Links

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

Crossrefs

Cf. A000961, A246547, A246549, A168363.

Programs

Maple

N:= 10^7: # to get all terms <= N
{seq(seq(p^m, m=4..floor(log[p](N))), p = select(isprime,[2,seq(2*i+1,i=1..floor(N^(1/4)))]))}; # Robert Israel, Aug 29 2014

Mathematica

With[{max = 10^6}, Sort @ Flatten @ Table[p^Range[4, Floor[Log[p, max]]], {p, Select[Range[Surd[max, 4]], PrimeQ]}]] (* Amiram Eldar, Oct 24 2020 *)

PARI

m=10^7; v=[]; forprime(p=2, m^(1/4), e=4; while(p^e<=m, v=concat(v, p^e); e++)); v=vecsort(v) \\ Jens Kruse Andersen, Aug 29 2014

Python

from math import isqrt
from sympy import primerange, integer_nthroot, primepi
def A246550(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(4, 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 12 2024

Formula

Sum_{n>=1} 1/a(n) = Sum_{p prime} 1/(p^3*(p-1)) = 0.1461466097... - Amiram Eldar, Oct 24 2020