A070265 Odd powers: numbers n = m^e with e > 1 odd.

1, 8, 27, 32, 64, 125, 128, 216, 243, 343, 512, 729, 1000, 1024, 1331, 1728, 2048, 2187, 2197, 2744, 3125, 3375, 4096, 4913, 5832, 6859, 7776, 8000, 8192, 9261, 10648, 12167, 13824, 15625, 16384, 16807, 17576, 19683, 21952, 24389, 27000

Offset: 1

Author

Eric W. Weisstein, May 07 2002

Keywords

• nonn
• easy

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Odd Powers.

Crossrefs

Cf. A001597, A008683, A097054.

Programs

Maple

N:= 10^6: # to get all terms <= N
{1,seq(seq(a^(2*k+1), k = 1 .. floor((log[a](N)-1)/2)),a=2..floor(N^(1/3)))};
# if using Maple 11 or earlier, uncomment the next line
# sort(convert(%,list)); # Robert Israel, Apr 24 2015

Mathematica

nn = 27000; Join[{1}, Union[Flatten[Table[n^i, {i, Prime[Range[2, PrimePi[Log[2, nn]]]]}, {n, 2, nn^(1/i)}]]]] (* T. D. Noe, Apr 19 2011 *)

PARI

is(x)=p=ispower(x);x==1||(p>1&bitand(p,p-1)!=0) \\ Charles R Greathouse IV, Apr 20 2015; corrected by Jeppe Stig Nielsen, Jul 14 2015

PARI

list(lim)=my(v=List([1])); forstep(e=3,log(lim)\log(2),2, for(n=2,sqrtnint(lim\1,e), listput(v,n^e))); Set(v) \\ Charles R Greathouse IV, Apr 20 2015

Python

from sympy import mobius, integer_nthroot
def A070265(n):
    def f(x): return int(n-2+x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(3,x.bit_length(),2)))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 14 2024

Formula

a(n) ~ n^3. - Charles R Greathouse IV, Apr 20 2015

Sum_{n>=1} 1/a(n) = 1 + Sum_{k>=1} mu(2*k+1)*(1-zeta(2*k+1)) = 1.2479294392... - Amiram Eldar, Dec 21 2020

Extensions

Name clarified by Charles R Greathouse IV, Oct 16 2015