A391980 Achilles numbers divisible by at least 2 cubes greater than 1.

432, 648, 864, 1944, 2000, 2592, 3456, 3888, 4000, 5000, 5400, 5488, 6912, 9000, 10125, 10368, 10584, 10800, 10976, 13500, 15552, 16000, 16200, 16875, 17496, 18000, 19208, 20000, 21168, 21296, 21600, 23328, 24696, 25000, 26136, 27648, 27783, 30375, 31104, 31752

Offset

1,1

Comments

Intersection of A052486 and A391968.

Links

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

Index entries for sequences related to powerful numbers.

Formula

Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - Sum_{k>=2} mu(k)*(1-zeta(k)) - 1 - (15/Pi^2) * Sum_{p prime} p/(p^4-1) + Sum_{p prime} 1/(p*(p^2-1)) = 0.0115978711467134791335... . - Amiram Eldar, Jan 06 2026

Example

Table of n, a(n) for select n:
  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
 11      5400 = 2^3 * 3^3 * 5^2
 60     54000 = 2^4 * 3^3 * 5^3
618   2646000 = 2^4 * 3^3 * 5^3 * 7^2

Mathematica

Select[Range[2^16], And[Count[#, _?(# > 2 &)] > 1, AllTrue[#, # > 1 &], GCD @@ # == 1] &[FactorInteger[#][[;; , -1]] ] &] (* or *)
With[{nn = 2^16}, Select[Rest@ Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3] } ],  And[Count[#, _?(# > 2 &)] > 1, GCD @@ # == 1] &[FactorInteger[#][[;; , -1]] ] &]]

Prog

(PARI) isok(k) = {my(e = vecsort(factor(k)[, 2])); #e > 1 && e[1] > 1 && e[#e-1] > 2 && gcd(e) == 1; } \\ Amiram Eldar, Jan 06 2026

Crossrefs

Cf. Subsets: A388293.

Supersets: A001694, A013929, A024619, A052486, A126706, A286708, A391968.

Cf. A072102, A082020, A082695, A369632.

Sequence in context: A337670 A389551 A391593 * A388293 A392908 A231231

Adjacent sequences: A391977 A391978 A391979 * A391981 A391982 A391983

Keyword

nonn,easy

Author

Michael De Vlieger, Dec 26 2025

Status

approved