A391491 Powerful numbers that are both divisible by only one prime cube and have more than 1 distinct prime factor.

72, 108, 144, 200, 288, 324, 392, 400, 500, 576, 675, 784, 800, 968, 972, 1125, 1152, 1323, 1352, 1372, 1568, 1600, 1800, 1936, 2025, 2304, 2312, 2500, 2700, 2704, 2888, 2916, 3087, 3136, 3200, 3267, 3528, 3600, 3872, 3969, 4232, 4500, 4563, 4608, 4624, 5292, 5324, 5408

Offset

1,1

Comments

This sequence is A372404 \ A376936.

Numbers in this sequence are neither cubefree nor cubeful (in A362148), hence are in A389558, a proper subset of A286708.

A389558(48) = 5400 = 2^3 * 3^3 * 5, divisible by 2 prime cubes; a(48) = 5408.

Numbers whose prime factorization exponents include at least one 2, and only one exponent that is larger than 2. - Amiram Eldar, Dec 21 2025

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} 1/((p-1)*(p^2+1)) - Sum_{p prime} 1/((p-1)*p^2) = 0.081675144208728966934... . - Amiram Eldar, Dec 21 2025

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     144 = 2^4 * 3^2
  4     200 = 2^3 * 5^2
  5     288 = 2^5 * 3^2
  6     324 = 2^2 * 3^4
 11     675 = 3^3 * 5^2
 23    1800 = 2^3 * 3^2 * 5^2
 29    2700 = 2^2 * 3^3 * 5^2
 37    3528 = 2^3 * 3^2 * 7^2
 38    3600 = 2^4 * 3^2 * 5^2
241   88200 = 2^3 * 3^2 * 5^2 * 7^2

Mathematica

With[{nn = 5400}, Select[Rest@ Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}], And[Length[#] > 1, Count[#, _?(# > 2 &)] == 1] &@ FactorInteger[#][[;; , -1]] &] ]

Prog

(PARI) isok(k) = {my(e = vecsort(factor(k)[, 2])); #e > 1 && e[1] > 1 && e[#e-1] == 2 && e[#e] > 2; } \\ Amiram Eldar, Dec 21 2025

Crossrefs

Cf. A001694, A024619, A046099, A059404, A286708, A359280, A362148, A372404, A378767, A389558.

Cf. A082020, A152441.

Sequence in context: A359280 A395067 A389558 * A307758 A376720 A272191

Adjacent sequences: A391488 A391489 A391490 * A391492 A391493 A391494

Keyword

nonn,easy

Author

Michael De Vlieger, Dec 10 2025

Status

approved