A075109 Odd perfect powers (1 together with numbers m^k, m odd, k >= 2).

1, 9, 25, 27, 49, 81, 121, 125, 169, 225, 243, 289, 343, 361, 441, 529, 625, 729, 841, 961, 1089, 1225, 1331, 1369, 1521, 1681, 1849, 2025, 2187, 2197, 2209, 2401, 2601, 2809, 3025, 3125, 3249, 3375, 3481, 3721, 3969, 4225, 4489, 4761, 4913, 5041, 5329, 5625

Offset: 1

Author

Zak Seidov, Oct 11 2002

Keywords

• easy
• nonn

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Intersection of A001597 and A005408.

Cf. A008683, A075090.

Programs

Haskell

a075109 n = a075109_list !! (n-1)
a075109_list = filter odd a001597_list  -- Reinhard Zumkeller, Oct 04 2012

Magma

[1] cat [n : n in [3..6000 by 2] | IsPower(n) ]; // Vincenzo Librandi, Mar 31 2014

Maple

q:= n-> n=1 or n::odd and igcd(seq(i[2], i=ifactors(n)[2]))>1:
select(q, [$1..6000])[];  # Alois P. Heinz, May 04 2022

Mathematica

Take[Union[Flatten[Table[a^b, {a, 1, 99, 2}, {b, 2, 15}]]], 50] (* Alonso del Arte, Nov 22 2011 *)

PARI

isok(m) = (m==1) || ((m%2) && ispower(m)); \\ Michel Marcus, May 04 2022

Python

from sympy import mobius, integer_nthroot
def A075109(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n-1+x+sum(mobius(k)*((integer_nthroot(x,k)[0]+1>>1)-1) for k in range(2,x.bit_length())))
    return bisection(f,n,n) # Chai Wah Wu, Feb 25 2025

Formula

Sum_{n>=1} 1/a(n) = 1 + Sum_{k>=2} mu(k)*(1-zeta(k)*(2^k-1)/2^k) = 1.2890375574... - Amiram Eldar, Dec 19 2020

Extensions

Definition clarified by N. J. A. Sloane, Dec 25 2021