A389959 Powers k^m, m > 0, of Achilles numbers k.

72, 108, 200, 288, 392, 432, 500, 648, 675, 800, 864, 968, 972, 1125, 1152, 1323, 1352, 1372, 1568, 1800, 1944, 2000, 2312, 2592, 2700, 2888, 3087, 3200, 3267, 3456, 3528, 3872, 3888, 4000, 4232, 4500, 4563, 4608, 5000, 5184, 5292, 5324, 5400, 5408, 5488, 6075

Offset

1,1

Comments

Union of A383394 and A052486, where the former contains the perfect powers of the latter.

Analogous to A182853, union of A303606 and A120944.

A059404 is the union of this sequence and A390055.

A024619 is the union of this sequence, A390055, and A182853.

A013929 is the union of this sequence, A390055, and A072777.

A286708 is the union of this sequence, A303606, and A386762.

A001694\{1} is the union of this sequence, A303606, A386762, and A246547.

Proper subset of A126706.

Links

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

Index entries for sequences related to powerful numbers.

Example

Table of n, a(n) for select n:
  n   a(n)
----------------------------
 1      72 = 2^3 * 3^2
 2     108 = 2^2 * 3^3
 3     200 = 2^3 * 5^2
 4     288 = 2^5 * 3^2
 5     392 = 2^3 * 7^2
 6     432 = 2^4 * 3^3
 7     500 = 2^2 * 5^3
 8     648 = 2^3 * 3^4
 9     675 = 3^3 * 5^2
20    1800 = 2^3 * 3^2 * 5^2
40    5184 = 2^6 * 3^4       =  72^2
69   11664 = 2^4 * 3^6       = 108^2

Mathematica

nn = 2^13; i = k = 1; MapIndexed[Set[S[First[#2]], #1] &, Rest@ Select[Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}], GCD @@ FactorInteger[#][[;; , -1]] == 1 &] ]; Union@ Reap[While[j = 1; While[S[i]^j < nn, Sow[S[i]^j]; j++]; j > 1, k++; i++] ][[-1, 1]]

Prog

(Python)
from math import isqrt
from sympy import integer_nthroot, mobius
from oeis_sequences.OEISsequences import bisection, squarefreepi
def A389959(n):
    def g(x):
        c, l = squarefreepi(integer_nthroot(x, 3)[0])+sum(mobius(k)*(integer_nthroot(x, k)[0]-1) for k in range(2, x.bit_length()))-1, 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2, 3)[0]+1
            w = squarefreepi(k2-1)
            c += j*(w-l)
            l, j = w, isqrt(x//k2**3)
        return c-l
    def f(x): return n+x-g(x)-sum(g(integer_nthroot(x, k)[0]) for k in range(2, x.bit_length()))
    return bisection(f, n, n) # Chai Wah Wu, Oct 31 2025

Crossrefs

Cf. A001694, A013929, A024619, A052486, A059404, A072777, A126706, A182853, A246547, A286708, A303606, A383394, A386762, A390055.

Sequence in context: A376720 A272191 A072412 * A052486 A393816 A393941

Adjacent sequences: A389956 A389957 A389958 * A389960 A389961 A389962

Keyword

nonn

Author

Michael De Vlieger, Oct 23 2025

Status

approved