A384518 Nonsquarefree numbers that are squarefree numbers raised to an odd power.

8, 27, 32, 125, 128, 216, 243, 343, 512, 1000, 1331, 2048, 2187, 2197, 2744, 3125, 3375, 4913, 6859, 7776, 8192, 9261, 10648, 12167, 16807, 17576, 19683, 24389, 27000, 29791, 32768, 35937, 39304, 42875, 50653, 54872, 59319, 68921, 74088, 78125, 79507, 97336, 100000

Offset: 1

Author

Amiram Eldar, Jun 01 2025

Keywords

• nonn

Comments

Subsequence of A097054 and first differs from it at n = 12: A097054(12) = 1728 = 2^6 * 3^3 is not a term of this sequence.

Numbers whose prime factorization exponents are equal, odd and larger than 1.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from exponents in factorization of n.
• Index entries for sequences related to powerful numbers.

Crossrefs

Intersection of A072777 and A268335.

Equals A072777 \ A384517.

Subsequence of A097054.

Cf. A005117.

Programs

Mathematica

Select[Range[10^5], Length[(u = Union[FactorInteger[#][[;; , 2]]])] == 1 && u[[1]] > 1 && OddQ[u[[1]]] &]

PARI

isok(k) = {my(s, e = ispower(k, , &s)); e % 2 && issquarefree(s);}

Python

from math import isqrt
from sympy import mobius, integer_nthroot
def A384518(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 g(x): return sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    def f(x): return n+x-sum(g(integer_nthroot(x,e)[0])-1 for e in range(3,x.bit_length(),2))
    return bisection(f,n,n) # Chai Wah Wu, Jun 01 2025

Formula

Sum_{n>=1} 1/a(n) = Sum_{k>=1} (zeta(2*k+1)/zeta(4*k+2)-1) = 0.22841193284408713846... .