A390435 Squares of Achilles numbers.

5184, 11664, 40000, 82944, 153664, 186624, 250000, 419904, 455625, 640000, 746496, 937024, 944784, 1265625, 1327104, 1750329, 1827904, 1882384, 2458624, 3240000, 3779136, 4000000, 5345344, 6718464, 7290000, 8340544, 9529569, 10240000, 10673289, 11943936, 12446784

Offset

1,1

Comments

Intersection of A000290 and A383394.

A383394 is the union of this sequence and A390436, disjoint sets.

A389959 is the union of this sequence, A052486, and A390436, disjoint sets.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Index entries for sequences related to powerful numbers.

Formula

a(n) = A052486(n)^2.

From Amiram Eldar, Nov 07 2025: (Start)

Equals A374291 \ A340588.

Sum_{n>=1} 1/a(n) = zeta(4)*zeta(6)/zeta(12) - 1 - Sum_{k>=2} mu(k)*(1-zeta(2*k)) = 0.00034780766944431258... . (End)

Example

Table of n, a(n) for select n:
 n      a(n)
---------------------------------------
 1     5184 =   72^2 = 2^6  * 3^4
 2    11664 =  108^2 = 2^4  * 3^6
 3    40000 =  200^2 = 2^6  * 5^4
 4    82944 =  288^2 = 2^10 * 3^4
 5   153664 =  392^2 = 2^6  * 7^4
 6   186624 =  432^2 = 2^8  * 3^6
 7   250000 =  500^2 = 2^4  * 5^6
 8   419904 =  648^2 = 2^6  * 3^8
 9   455625 =  675^2 = 3^6  * 5^4
10   640000 =  800^2 = 2^10 * 5^4
11   746496 =  864^2 = 2^10 * 3^6
20  3240000 = 1800^2 = 2^6  * 3^4 * 5^4

Mathematica

nn = 2^12; Rest[Select[Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}], GCD @@ FactorInteger[#][[;; , -1]] == 1 &]]^2

Prog

(Python)
from math import isqrt
from sympy import integer_nthroot, mobius
from oeis_sequences.OEISsequences import bisection, squarefreepi
def A390435(n):
    def f(x):
        y = isqrt(x)
        c, l = n+x+1, 0
        j = isqrt(y)
        while j>1:
            k2 = integer_nthroot(y//j**2, 3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(y//k2**3)
        c -= squarefreepi(integer_nthroot(y, 3)[0])-l+sum(mobius(k)*(integer_nthroot(y, k)[0]-1) for k in range(2, y.bit_length()))
        return c
    return bisection(f, n, n) # Chai Wah Wu, Nov 06 2025
(PARI) isok(k) = k > 1 && issquare(k) && ispowerful(sqrtint(k)) && !ispower(sqrtint(k)); \\ Amiram Eldar, Nov 07 2025

Crossrefs

Cf. A000290, A052486, A383394, A389959, A390436.

Cf. A001597, A001694, A013929, A024619, A126706, A131605, A286708, A340588, A374291.

Sequence in context: A061050 A383394 A391376 * A175745 A190464 A389226

Adjacent sequences: A390432 A390433 A390434 * A390436 A390437 A390438

Keyword

nonn,easy

Author

Michael De Vlieger, Nov 05 2025

Status

approved