A391923 Achilles numbers divisible by only 1 cube greater than 1.

72, 108, 200, 288, 392, 500, 675, 800, 968, 972, 1125, 1152, 1323, 1352, 1372, 1568, 1800, 2312, 2700, 2888, 3087, 3200, 3267, 3528, 3872, 4232, 4500, 4563, 4608, 5292, 5324, 5408, 6075, 6125, 6272, 6728, 7200, 7688, 7803, 8575, 8712, 8748, 8788, 9248, 9747, 9800

Offset

1,1

Comments

Numbers of the form p^m * k^2, where m > 2 is odd, prime p is coprime to k, and k > 1 is squarefree.

Proper subset of A390539 (noncubefull Achilles numbers); A390539(32) = 5400 = 2^3 * 3^3 * 5 is not in this sequence.

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) = (15/Pi^2) * Sum_{p prime} p/(p^4-1) - Sum_{p prime} 1/(p*(p^2-1)) = 0.057534197269100859229... . - Amiram Eldar, Jan 04 2026

Example

Table of n, a(n) for select n:
  n     a(n)
--------------------------------------------------
  1      72 = 2^3 *  3^2
  2     108 = 2^2 *  3^3
  3     200 = 2^3 *  5^2
  4     288 = 2^5 *  3^2
  5     392 = 2^3 *  7^2
  6     500 = 2^2 *  5^3
  7     675 = 3^3 *  5^2
  8     800 = 2^5 *  5^2
  9     968 = 2^3 * 11^2
 10     972 = 2^2 *  3^5
 17    1800 = 2^3 *  3^2 * 5^2       = 2^3 *  15^2
155   88200 = 2^3 *  3^2 * 5^2 * 7^2 = 2^3 * 105^2

Mathematica

Select[Range[10^4], And[Count[#, _?(# > 2 &)] < 2, AllTrue[#, # > 1 &], GCD @@ # == 1] &[FactorInteger[#][[;; , -1]] ] &] (* or *)
With[{nn = 10^4},  Select[Rest@  Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3] }, {a, Sqrt[nn/b^3] } ], And[Count[#, _?(# > 2 &)] < 2, GCD @@ # == 1] &[FactorInteger[#][[;; , -1]] ] &] ]

Prog

(PARI) isok(k) = if(k < 2, 0, my(e = vecsort(factor(k)[, 2])); #e > 1 && e[1] == 2 && e[#e-1] == 2 && e[#e] % 2); \\ Amiram Eldar, Jan 04 2026

Crossrefs

Supersets A001694, A013929, A024619, A052486, A126706, A286708, A390539, A391968.

Cf. A082020, A369632.

Sequence in context: A378859 A114128 A390539 * A375074 A375143 A375073

Adjacent sequences: A391920 A391921 A391922 * A391924 A391925 A391926

Keyword

nonn,easy

Author

Michael De Vlieger, Dec 25 2025

Status

approved