The OEIS is supported by the many generous donors to the OEIS Foundation.
%I #11 Jan 06 2026 11:11:14
%S 432,648,864,1944,2000,2592,3456,3888,4000,5000,5400,5488,6912,9000,
%T 10125,10368,10584,10800,10976,13500,15552,16000,16200,16875,17496,
%U 18000,19208,20000,21168,21296,21600,23328,24696,25000,26136,27648,27783,30375,31104,31752
%N Achilles numbers divisible by at least 2 cubes greater than 1.
%C Intersection of A052486 and A391968.
%H Michael De Vlieger, <a href="/A391980/b391980.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.
%F 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
%e Table of n, a(n) for select n:
%e n a(n)
%e -------------------------------------
%e 1 432 = 2^4 * 3^3
%e 2 648 = 2^3 * 3^4
%e 3 864 = 2^5 * 3^3
%e 4 1944 = 2^3 * 3^5
%e 5 2000 = 2^4 * 5^3
%e 6 2592 = 2^5 * 3^4
%e 7 3456 = 2^7 * 3^3
%e 8 3888 = 2^4 * 3^5
%e 9 4000 = 2^5 * 5^3
%e 11 5400 = 2^3 * 3^3 * 5^2
%e 60 54000 = 2^4 * 3^3 * 5^3
%e 618 2646000 = 2^4 * 3^3 * 5^3 * 7^2
%t Select[Range[2^16], And[Count[#, _?(# > 2 &)] > 1, AllTrue[#, # > 1 &], GCD @@ # == 1] &[FactorInteger[#][[;; , -1]] ] &] (* or *)
%t 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]] ] &]]
%o (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
%Y Cf. Subsets: A388293.
%Y Supersets: A001694, A013929, A024619, A052486, A126706, A286708, A391968.
%Y Cf. A072102, A082020, A082695, A369632.
%K nonn,easy
%O 1,1
%A _Michael De Vlieger_, Dec 26 2025