A340674 Numbers of the form s^(2^e), where s is a squarefree number, and e >= 1.

4, 9, 16, 25, 36, 49, 81, 100, 121, 169, 196, 225, 256, 289, 361, 441, 484, 529, 625, 676, 841, 900, 961, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1681, 1764, 1849, 2116, 2209, 2401, 2601, 2809, 3025, 3249, 3364, 3481, 3721, 3844, 4225, 4356, 4489, 4761, 4900

Offset: 1

Author

Antti Karttunen, Jan 31 2021

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A000079, A005117, A050376, A209229, A302777.

Positions of terms larger than 2 in A340673 (also in A340675), and of terms larger than 1 in A340676.

Subsequence of A072777 and of A340682.

Programs

Mathematica

Select[Range[10^4], Length[(u = Union[FactorInteger[#][[;; , 2]]])] == 1 && u[[1]] > 1 && u[[1]] == 2^IntegerExponent[u[[1]], 2] &] (* Amiram Eldar, Feb 08 2021 *)

PARI

A209229(n) = (n && !bitand(n,n-1));
isA340674(n) = { my(b,e); (((e=ispower(n,,&b))>0)&&issquarefree(b)&&A209229(e)); };

Python

from math import isqrt
from sympy import mobius, integer_nthroot
def A340674(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,1<Chai Wah Wu, Jun 01 2025

Formula

Sum_{n>=1} 1/a(n) = Sum_{k>=1} (zeta(2^k)/zeta(2^(k+1))-1) = 0.6018231854... - Amiram Eldar, Feb 08 2021