A391755 Cubefull numbers with more than 2 distinct prime factors.

27000, 54000, 74088, 81000, 108000, 135000, 148176, 162000, 216000, 222264, 243000, 270000, 287496, 296352, 324000, 343000, 405000, 432000, 444528, 474552, 486000, 518616, 540000, 574992, 592704, 648000, 666792, 675000, 686000, 729000, 810000, 862488, 864000, 889056

Offset

1,1

Comments

Intersection of A036966 (cubefull numbers) and A000977 (numbers with more than 2 distinct prime factors).

Proper subset of A390950 which is the intersection of A001694 (powerful numbers) and A000977.

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) = Product_{p prime}(1 + 1/(p^2*(p-1))) - Sum_{p prime} 1/(p^2*(p-1)) - 1 - ((Sum_{p prime} (1/(p^2*(p-1))))^2 - Sum_{p prime} (1/(p^4*(p-1)^2)))/2 = 0.00023303003383679282415... . - Amiram Eldar, Dec 23 2025

Example

Table of n, a(n) for select n:
 n        a(n)
-------------------------------------
  1     27000 = 2^3 * 3^3 * 5^3
  2     54000 = 2^4 * 3^3 * 5^3
  3     74088 = 2^3 * 3^3 * 7^3
  4     81000 = 2^3 * 3^4 * 5^3
  5    108000 = 2^5 * 3^3 * 5^3
  6    135000 = 2^3 * 3^3 * 5^4
  7    148176 = 2^4 * 3^3 * 7^3
  8    162000 = 2^4 * 3^4 * 5^3
  9    216000 = 2^6 * 3^3 * 5^3
 10    222264 = 2^3 * 3^4 * 7^3
 41   1157625 = 3^3 * 5^3 * 7^3
161   9261000 = 2^3 * 3^3 * 5^3 * 7^3

Mathematica

nn = 2^20; s = 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]}]; Select[s, PrimeNu[#] > 2 &]

Crossrefs

Cf. A000977, A001694, A036966, A126706, A286708, A390950.

Sequence in context: A269115 A249496 A227348 * A386802 A176359 A162144

Adjacent sequences: A391752 A391753 A391754 * A391756 A391757 A391758

Keyword

nonn,easy

Author

Michael De Vlieger, Dec 22 2025

Status

approved