A389227 Numbers in A126706 that are less than the square of their squarefree kernel.

12, 18, 20, 24, 28, 40, 44, 45, 50, 52, 56, 60, 63, 68, 75, 76, 80, 84, 88, 90, 92, 98, 99, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 198, 204, 207, 208, 212, 220, 228

Offset

1,1

Comments

Intersection of S and T, where S is any one of A126706, A059404, A303946, and A332785 and T is either A341646 or A390528.

A126706 is the union of this sequence and A370266, disjoint sets.

Union of disjoint sets A367708 and A364996 = union of disjoint sets A177492 and A370409.

Define sequence R(k) to be the set {m*k : rad(m) | k, m >= 1} for composite squarefree k (i.e., k in A120944). Then S(r) = r * {m : rad(m) | r} and so we have all terms in S(r) that do not exceed r^2 in this sequence. This is to say, given R(k,j) is the j-th term in R(k), that this sequence contains R(k,j) for 1 < j <= A010846(k)-1. As a consequence, this sequence contains no squarefree numbers, therefore occurs in the intersection of A013929 and A024619 = A126706.

The cardinality of the intersection of this sequence and R(k) is A010846(k)-2.

Links

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

Example

For k = 6, this sequence contains R(k,j) = A033485(j), j > A010846(6), i.e., {12, 18, 24}. Since 6 is the smallest term in A120944, a(1) = 12.
For k = 10, this sequence contains R(k,j) = A033486(j), j > A010846(10), i.e., {20, 40, 50, 80}. Therefore, a(3) = 20 is the first term in this sequence that is not in A033845.

Mathematica

Select[Range[240], And[Nor[PrimePowerQ[#1], SquareFreeQ[#1]], #1/#2 < #2] & @@ {#1, Times @@ FactorInteger[#][[;; , 1]]} &]

Crossrefs

Cf. A000290, A007947, A059404, A126706, A177492, A303946, A332785, A364996, A367708, A370266, A370409.

Sequence in context: A345381 A328930 A392537 * A364996 A391026 A363082

Adjacent sequences: A389224 A389225 A389226 * A389228 A389229 A389230

Keyword

nonn

Author

Michael De Vlieger, Nov 10 2025

Status

approved