A388293 Cubefull Achilles numbers.

432, 648, 864, 1944, 2000, 2592, 3456, 3888, 4000, 5000, 5488, 6912, 10125, 10368, 10976, 15552, 16000, 16875, 17496, 19208, 20000, 21296, 23328, 25000, 27648, 27783, 30375, 31104, 32000, 34992, 35152, 41472, 42592, 43904, 50000, 52488, 54000, 55296, 62208, 64827

Offset

1,1

Comments

A372695 is the union of this sequence, A383394, A388304, and A388549.

A036966 = A246549 U A372695, where A246549 is the intersection of A036966 and A000961, and A372695 is the intersection of A036966 and A024619.

The cubefull numbers are the disjoint union of this sequence, A076467, and the terms of A052486 squared. - Amiram Eldar, Oct 01 2025

Links

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

Index entries for sequences related to powerful numbers.

Formula

Intersection of A036966 and A052486.

Sum_{n>=1} 1/a(n) = Product_{p prime}(1 + 1/(p^2*(p-1))) - zeta(4)*zeta(6)/zeta(12) - 1 + zeta(2) + Sum_{k>=2} mu(k)*(zeta(k)-1) = 0.0094307171480903586466... . - Amiram Eldar, Oct 01 2025

Example

 n    a(n)
---------------------
 1     432 = 2^4 * 3^3
 2     648 = 2^3 * 3^4
 3     864 = 2^5 * 3^3
 4    1944 = 2^3 * 3^5
 5    2000 = 2^4 * 5^3
 6    2592 = 2^5 * 3^4
 7    3456 = 2^7 * 3^3
 8    3888 = 2^4 * 3^5
 9    4000 = 2^5 * 5^3
10    5000 = 2^3 * 5^4
37   54000 = 2^4 * 3^3 * 5^3
46   81000 = 2^3 * 3^4 * 5^3

Mathematica

With[{nn = 2^16}, Select[Rest@ Union@ Flatten@ Table[a^5*b^4*c^3, {c, Surd[nn, 3]}, {b, Surd[nn/c^3, 4]}, {a, Surd[nn/(b^4*c^3), 5]}], GCD @@ FactorInteger[#][[All, -1]] == 1 &]]
(* Alternative: *)
isA388293[n_Integer] := Module[{factors, exponents},
  factors = FactorInteger[n];
  exponents = factors[[All, 2]];
  If[AnyTrue[exponents, # < 3 &], Return[False]];
  Return[GCD @@ exponents === 1]]
Select[Range[1, 7000], isA388293]  (* Peter Luschny, Dec 02 2025 *)

Prog

(Python)
from math import isqrt, gcd
from sympy import mobius, integer_nthroot, factorint
from oeis_sequences.OEISsequences import bisection, squarefreepi
def A388293(n):
    def f(x):
        y = isqrt(x)
        c = n+x-sum(mobius(k)*(integer_nthroot(x, k)[0]-1) 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()))
        for w in range(1, integer_nthroot(x, 5)[0]+1):
            if all(d<=1 for d in factorint(w).values()):
                for y in range(1, integer_nthroot(z:=x//w**5, 4)[0]+1):
                    if gcd(w, y)==1 and all(d<=1 for d in factorint(y).values()):
                        c -= integer_nthroot(z//y**4, 3)[0]
        return c
    return bisection(f, n, n) # Chai Wah Wu, Dec 01 2025
(SageMath)
def isA388293(n: int) -> bool:
    factors = factor(n)
    exps = [e for (_, e) in factors]
    if any(e < 3 for e in exps): return False
    return reduce(gcd, exps) == 1
print([n for n in range(2, 7000) if isA388293(n)])  # Peter Luschny, Dec 02 2025
(PARI) A388293(n)  = if(n<=1, 0, (e->(1==gcd(e) && vecmin(e)>2))(factor(n)[, 2])); \\ Antti Karttunen, Jan 31 2026

Crossrefs

Cf. A036966, A052486, A076467, A126706, A131605, A246549, A286708, A383394, A386762, A372695, A388304.

Sequence in context: A389551 A391593 A391980 * A392908 A231231 A179666

Adjacent sequences: A388290 A388291 A388292 * A388294 A388295 A388296

Keyword

nonn,easy

Author

Michael De Vlieger, Sep 20 2025

Status

approved