A390949 Nonsquarefree weak numbers that have more than 2 distinct prime factors.

60, 84, 90, 120, 126, 132, 140, 150, 156, 168, 180, 198, 204, 220, 228, 234, 240, 252, 260, 264, 270, 276, 280, 294, 300, 306, 308, 312, 315, 336, 340, 342, 348, 350, 360, 364, 372, 378, 380, 396, 408, 414, 420, 440, 444, 450, 456, 460, 468, 476, 480, 490, 492

Offset

1,1

Comments

Proper subset of A388305, since A388305 contains powerful numbers like 1800, 2700, and 3528. A388305(311) = 1800 = 2^3 * 3^2 * 5^2, while a(311) = 1804 = 2^2 * 11 * 41.

This sequence is { k : omega(k) > 2, k != rad(k), rad(k)^2 does not divide k }, where omega = A001221 and rad = A007947.

Let Q be the set of prime power factor exponents of k. In contrast, A388305 is the set of k such that card(Q) > 1. For this sequence, both card(Q) > 1 and 1 is an element of Q. The latter is not a condition for A388305.

Links

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

Formula

Intersection of A000977, A013929, and A052485.

Intersection of A375055 and A332785.

A332785 is the disjoint union of this sequence and A345381.

Example

Table of n, a(n) for select n:
 n    a(n)
------------------------------
 1    60 = 2^2 * 3   *  5
 2    84 = 2^2 * 3   *  7
 3    90 = 2   * 3^2 *  5
 4   120 = 2^3 * 3   *  5
 5   126 = 2   * 3^2 *  7
 6   132 = 2^2 * 3   * 11
 7   140 = 2^2 * 5   *  7
 8   150 = 2   * 3   *  5^2
 9   156 = 2^2 * 3   * 13
29   315 = 3^2 * 5   *  7
43   420 = 2^2 * 3   *  5 * 7
59   525 = 3   * 5^2 *  7

Mathematica

Select[Range[5000], And[2 < #1 < #2, #3 == 1] & @@ {Length[#], Total[#], Min[#]} &[FactorInteger[#][[;; , -1]] ] &]

Crossrefs

Cf. A000977, A013929, A052485, A332785, A375055, A388305.

Sequence in context: A261375 A349758 A217740 * A375055 A388305 A123712

Adjacent sequences: A390946 A390947 A390948 * A390950 A390951 A390952

Keyword

nonn

Author

Michael De Vlieger, Dec 02 2025

Status

approved