A387255 Cubefull powers k^m, m > 1, of composite numbers k that are not powerful.

216, 1000, 1296, 1728, 2744, 3375, 5184, 5832, 7776, 8000, 9261, 10000, 10648, 11664, 13824, 17576, 20736, 21952, 27000, 35937, 38416, 39304, 40000, 42875, 46656, 50625, 54872, 59319, 64000, 74088, 82944, 85184, 91125, 97336, 100000, 104976, 110592, 125000, 132651, 140608

Offset

1,1

Comments

Cubefull powers k^m, m > 1, such that k is in A106543; intersection of A036966 and A131605.

Disjoint union of A388304 (cubefull powers of squarefree composites A120944), A383394 (proper powers of Achilles numbers A052486), and A388549 (cubefull powers of nonsquarefree weak numbers A332785).

A076467 is the disjoint union of this sequence and A246549.

A131605 is the disjoint union of this sequence and A387254.

A372695 is the disjoint union of this sequence and A388293.

Squares in this sequence are confined to A383394.

Links

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

Index entries for sequences related to powerful numbers.

Example

Let r = A388304, let s = A388549, and let t = A383394.
Table of n, a(n) for n = 1..13:
 n     a(n)
-------------------------------------
 1     216 =  6^3 = 2^3 * 3^3  = r(1)
 2    1000 = 10^3 = 2^3 * 5^3  = r(2)
 3    1296 =  6^4 = 2^4 * 3^4  = r(3)
 4    1728 = 12^3 = 2^6 * 3^3  = s(1)
 5    2744 = 14^3 = 2^3 * 7^3  = r(4)
 6    3375 = 15^3 = 3^3 * 5^3  = r(5)
 7    5184 = 72^2 = 2^6 * 3^4  = t(1)
 8    5832 = 18^3 = 2^3 * 3^6  = s(2)
 9    7776 =  6^5 = 2^5 * 3^5  = r(6)
10    8000 = 20^3 = 2^6 * 5^3  = s(3)
11    9261 = 21^3 = 3^3 * 7^3  = r(7)
12   10000 = 10^4 = 2^4 * 5^4  = r(8)
13   10648 = 22^3 = 2^3 * 11^3 = r(9)

Mathematica

With[{nn = 150000}, Union@ Flatten@ Table[If[! PrimePowerQ[#], If[GCD @@ FactorInteger[#][[;; , -1]] > 1, #, Nothing], Nothing] &[a^5 * b^4 * c^3], {c, Surd[nn, 3]}, {b, Surd[nn/(c^3), 4]}, {a, Surd[nn/(b^4 * c^3), 5] } ] ]

Prog

(Python)
from math import isqrt
from sympy import mobius, integer_nthroot, primepi, factorint
from oeis_sequences.OEISsequences import bisection
def A387255(n):
    def f(x):
        y = isqrt(x)
        return int(n+1+x+sum(mobius(k)*((a:=integer_nthroot(x, k)[0])-1)+primepi(a) for k in range(3, x.bit_length()))-sum(isqrt(y//k**3) for k in range(1, integer_nthroot(y, 3)[0]+1) if all(d<=1 for d in factorint(k).values())))
    return bisection(f, n, n) # Chai Wah Wu, Dec 01 2025

Crossrefs

Cf. A001597, A001694, A036966, A052486, A076467, A106543, A120944, A126706, A131065, A246549, A286708, A332785, A372695, A383394, A387254, A388293, A388304, A388549.

Sequence in context: A250137 A386801 A109399 * A390905 A388304 A384006

Adjacent sequences: A387252 A387253 A387254 * A387256 A387257 A387258

Keyword

nonn,easy

Author

Michael De Vlieger, Nov 20 2025

Extensions

Missing term 7776 added by Chai Wah Wu, Dec 01 2025

Status

approved