A390950 Powerful numbers with more than 2 distinct prime factors.

900, 1764, 1800, 2700, 3528, 3600, 4356, 4500, 4900, 5292, 5400, 6084, 7056, 7200, 8100, 8712, 9000, 9800, 10404, 10584, 10800, 11025, 12100, 12168, 12348, 12996, 13068, 13500, 14112, 14400, 15876, 16200, 16900, 17424, 18000, 18252, 19044, 19600, 20808, 21168

Offset

1,1

Comments

A286708 is the disjoint union of this sequence and A355462.

Links

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

Formula

Intersection of A000977 and A001694.

Intersection of A286708 and A375055.

Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - Sum_{p prime} 1/(p*(p-1)) - 1 - ((Sum_{p prime} (1/(p*(p-1))))^2 - Sum_{p prime} (1/(p^2*(p-1)^2)))/2 = 0.012053688667932126846... . - Amiram Eldar, Dec 10 2025

Example

Table of n, a(n) for select n:
 n     a(n)
-----------------------------------
 1     900 = 2^2 * 3^2 *  5^2
 2    1764 = 2^2 * 3^2 *  7^2
 3    1800 = 2^3 * 3^2 *  5^2
 4    2700 = 2^2 * 3^3 *  5^2
 5    3528 = 2^3 * 3^2 *  7^2
 6    3600 = 2^4 * 3^2 *  5^2
 7    4356 = 2^2 * 3^2 * 11^2
 8    4500 = 2^2 * 3^2 *  5^3
 9    4900 = 2^2 * 5^2 *  7^2
10    5292 = 2^2 * 3^3 *  7^2
22   11025 = 3^2 * 5^2 *  7^2
79   44100 = 2^2 * 3^2 *  5^2 * 7^2

Mathematica

With[{nn = 25000}, Union@ Flatten@ Table[If[PrimeNu[#] > 2, #, Nothing] &[a^2*b^3], {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3] } ] ]

Prog

(PARI) is_a390950(n) = ispowerful(n) && omega(n)>2 \\ Hugo Pfoertner, Dec 07 2025

Crossrefs

Cf. A000977, A001694, A286708, A355462, A375055.

Sequence in context: A127658 A318720 A338540 * A137490 A129575 A328136

Adjacent sequences: A390947 A390948 A390949 * A390951 A390952 A390953

Keyword

nonn,easy

Author

Michael De Vlieger, Dec 01 2025

Status

approved