A384517 Nonsquarefree numbers that are squarefree numbers raised to an even power.

4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 169, 196, 225, 256, 289, 361, 441, 484, 529, 625, 676, 729, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1681, 1764, 1849, 2116, 2209, 2401, 2601, 2809, 3025, 3249, 3364, 3481, 3721, 3844, 4096, 4225, 4356

Offset: 1

Author

Amiram Eldar, Jun 01 2025

Keywords

• nonn

Comments

Differs from its subsequence A340674 by having the terms 64, 729, 1024, 4096, .... .

Numbers whose prime factorization exponents are equal and even.

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 A000290 and A072777.

Equals A072777 \ A384518.

A340674 is a subsequence.

Cf. A005117, A062770, A072774, A231273, A231327.

Programs

Mathematica

Select[Range[2, 100], SameQ @@ FactorInteger[#][[;;, 2]] &]^2

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 A384517(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(2,x.bit_length(),2))
    return bisection(f,n,n) # Chai Wah Wu, Jun 01 2025

Formula

a(n) = A062770(n)^2 = A072774(n+1)^2.

Sum_{n>=1} 1/a(n) = Sum_{k>=1} (zeta(2*k)/zeta(4*k)-1) = Sum{k>=1} (A231327(k)/(A231273(k)*Pi^(2*k)) - 1) = 0.62022193512079649421... .