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

1,1

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.

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... .

Mathematica

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

Prog

(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

Crossrefs

Intersection of A000290 and A072777.

Equals A072777 \ A384518.

A340674 is a subsequence.

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

Sequence in context: A018885 A025741 A387918 * A179459 A325148 A325149

Adjacent sequences: A384514 A384515 A384516 * A384518 A384519 A384520

Keyword

nonn

Author

Amiram Eldar, Jun 01 2025

Status

approved