A389528 Numbers in A024619 that exceed the square of their squarefree kernel.

48, 54, 72, 96, 108, 144, 160, 162, 192, 200, 216, 224, 250, 288, 320, 324, 375, 384, 392, 400, 405, 432, 448, 486, 500, 567, 576, 640, 648, 675, 686, 704, 768, 784, 800, 832, 864, 896, 960, 968, 972, 1000, 1029, 1080, 1125, 1152, 1200, 1215, 1250, 1280, 1296

Offset

1,1

Comments

Numbers k with more than 1 distinct prime factor such that k > rad(k)^2, where rad = A007947.

Numbers k with more than 1 distinct prime factor such that A003557(k) > A007947(k), where A003557(k) = k/rad(k). This is tantamount to numbers k with more than 1 distinct prime factor such that A003557(k) > sqrt(k).

This sequence is { k : A001221(k) > 1, A003557(k) > A007947(k) } = A059172 \ A246549.

Intersection of A126706 and A059172.

Define sequence S(r) to be the set {m*r : rad(m) | r, m >= 1} for composite squarefree r (i.e., r in A120944). Then S(r) = r * {m : rad(m) | r} and so we have all terms in S(r) that exceed r^2 in this sequence. This is to say, given S(r,j) is the j-th term in S(r), that this sequence contains S(r,j) for j > A010846(r). As a consequence, this sequence contains no squarefree numbers, therefore occurs in the intersection of A013929 and A024619 = A126706.

A126706 is the union of this sequence and A370409.

Links

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

Example

For r = 6, this sequence contains S(r,j) = A033485(j), j > A010846(6), i.e., {48, 54, 72, 96, 108, ...}. Since 6 is the smallest term in A120944, a(1) = 48.
For r = 10, this sequence contains S(r,j) = A033486(j), j > A010846(10), i.e., {160, 200, 250, 320, 400, ...}. Therefore, a(7) = 160 is the first term in this sequence that is not in A033845.

Mathematica

Select[Range[2^16], And[#2 > 1, #1/#3 > #3] & @@ {#1, Length[#2], Apply[Times, #2]^2} & @@ {#, FactorInteger[#][[;; , 1]] } &]

Crossrefs

Cf. A003557, A007947, A010846, A013929, A024619, A059172, A120944, A126706, A246549, A370409.

Sequence in context: A328738 A067191 A080854 * A389752 A389753 A255267

Adjacent sequences: A389525 A389526 A389527 * A389529 A389530 A389531

Keyword

nonn

Author

Michael De Vlieger, Nov 09 2025

Status

approved