A387257 Cubefull powers k^m, m > 1, such that k is neither squarefree nor perfect power.

1728, 5184, 5832, 8000, 11664, 13824, 20736, 21952, 40000, 64000, 82944, 85184, 91125, 104976, 110592, 125000, 140608, 153664, 157464, 160000, 175616, 186624, 216000, 248832, 250000, 250047, 314432, 331776, 373248, 419904, 421875, 438976, 455625, 512000, 592704

Offset

1,1

Comments

Union of A383394 and A388549 = A076467 \ A390540 = A389551 \ A388293.

A389864 is the disjoint union of this sequence and A389947.

All squares in this sequence are in A390435, squares of Achilles numbers (A052486).

Links

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

Index entries for sequences related to powerful numbers.

Example

Let s = A388549 (cubefull numbers in A386762, i.e., perfect powers of numbers in A332785) and let t = A383394 (i.e., perfect powers of Achilles numbers).
Table of n, a(n) for select n:
 n      a(n)
---------------------------------------------
 1     1728 =  12^3 = 2^6  * 3^3       = s(1)
 2     5184 =  72^2 = 2^6  * 3^4       = t(1)
 3     5832 =  18^3 = 2^3  * 3^6       = s(2)
 4     8000 =  20^3 = 2^6  * 5^3       = s(3)
 5    11664 = 108^2 = 2^4  * 3^6       = t(2)
 6    13824 =  24^3 = 2^9  * 3^3       = s(4)
 7    20736 =  12^4 = 2^8  * 3^4       = s(5)
 8    21952 =  28^3 = 2^6  * 7^3       = s(6)
 9    40000 = 200^2 = 2^6  * 5^4       = t(3)
10    64000 =  40^3 = 2^9  * 5^3       = s(7)
11    82944 = 288^2 = 2^10 * 3^4       = t(4)
23   216000 =  60^3 = 2^6  * 3^3 * 5^3 = s(8)

Mathematica

nn = 600000; mm = Sqrt[nn]; i = 1; fQ[x_] := And[Max[#] > 1, GCD @@ # == 1] &@ FactorInteger[x][[;; , -1]]; MapIndexed[Set[S[First[#2]], #1] &, Select[Range@ Sqrt[nn], fQ]]; Union@ Reap[While[j = 2; While[S[i]^j < nn, If[AllTrue[FactorInteger[#][[;; , -1]], # > 2 &], Sow[#] ] &[S[i]^j]; j++]; j > 2, i++] ][[-1, 1]]

Prog

(Python)
from math import isqrt
from sympy import integer_nthroot, mobius
from oeis_sequences.OEISsequences import bisection, squarefreepi
def A387257(n):
    def f(x):
        c, l, y = n+x+1, 0, isqrt(x)
        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)
        return c+l+sum(mobius(k)*((z:=integer_nthroot(x, k)[0])-1)+(0 if k== 6 else squarefreepi(z))-1 for k in range(3, x.bit_length()))
    return bisection(f, n, n) # Chai Wah Wu, Dec 01 2025

Crossrefs

Cf. A001597, A001694, A036966, A052486, A076467, A126706, A131605, A286708, A332785, A383394, A386762, A388293, A388549, A389551, A390435, A390540.

Sequence in context: A032354 A305475 A202458 * A392098 A388549 A392564

Adjacent sequences: A387254 A387255 A387256 * A387258 A387259 A387260

Keyword

nonn

Author

Michael De Vlieger, Nov 19 2025

Status

approved