A389752 Numbers in A059404 that exceed the square of their squarefree kernel.

48, 54, 72, 96, 108, 144, 160, 162, 192, 200, 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, 1029, 1080, 1125, 1152, 1200, 1215, 1250, 1280, 1323, 1350, 1352

Offset

1,1

Comments

Numbers k > rad(k)^2 such that prime power factor p^m | k has more than one distinct m, where rad = A007947 is the squarefree kernel.

Intersection of A059404 and A059172.

Intersection of A059404 and A341645.

Union of the following disjoint subsets: A359280 and A366250, A389753 and A389864.

A389528 is the union of this sequence and A388304, disjoint sets.

A370266 is the union of this sequence and A303606, disjoint sets.

A059404 is the union of this sequence and A389227, disjoint sets.

A126706 is the union of this sequence, A389227, and A303606, disjoint sets.

Let R(k) = {m*k : rad(m) | k} with squarefree composite k > 1 (k in A120944). This sequence intersects R(k) for m > k such that m*k is not a perfect power of k, i.e., m*k not in A303606.

If R(k,j) is the j-th smallest term in R(k), then R(k,j) with j <= A010846(k) is not in this sequence. Therefore, s in R(k,j) \ A303606, j > A010846(k) are the terms in this sequence with rad(s) = k. In this manner, this sequence is the local complement of powers {k^m : m > 0} with respect to R(k,j), j > A010846(k), across squarefree composite k.

The number s = R(k, A010846(k)+1) is the smallest term in this sequence with rad(s) = k.

Links

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

Example

A059404(1) = 12 is not a term since 12 < 36.
For numbers with squarefree kernel 6 (in R(6) = A033845), the smallest number with this quality is R(6,6) = 48 = 2^3 * 3^1 which has more than 1 distinct prime power factor exponent, thus, a(1) = 48.
For numbers with squarefree kernel 10 (in R(10) = A033846), the smallest number with this quality is 160 = 2^4 * 5^1, which has more than 1 distinct prime power factor exponent, thus, a(7) = 160 is the smallest number in this sequence that does not have squarefree kernel 6.
The number 36 = 6^2 = 2^2 * 3^2 is not a term since it is the square of a squarefree number and thus its prime power factor exponents are the same. Furthermore, 36 = rad(36)^2.
The number 1000 = 10^3 = 2^3 * 5^3 is not a term since it is the square of a squarefree number and thus its prime power factor exponents are the same.

Mathematica

Select[Range[1500], And[! SquareFreeQ[#1], CountDistinct[#2] > 1, #1/#3 > #3] & @@ {#1, #2[[;; , -1]], Times @@ #2[[;; , 1]]} & @@ {#, FactorInteger[#]} &]

Crossrefs

Cf. A003557, A007947, A010846, A059404, A059172, A120944, A126706, A303606, A341645, A366250, A370266, A388304, A389227, A389528, A389864, A389753.

Sequence in context: A067191 A080854 A389528 * A389753 A255267 A366250

Adjacent sequences: A389749 A389750 A389751 * A389753 A389754 A389755

Keyword

nonn

Author

Michael De Vlieger, Nov 11 2025

Status

approved