A389753 Nonsquarefree numbers that are not perfect powers that exceed the square of their squarefree kernel.

48, 54, 72, 96, 108, 160, 162, 192, 200, 224, 250, 288, 320, 375, 384, 392, 405, 432, 448, 486, 500, 567, 640, 648, 675, 686, 704, 768, 800, 832, 864, 896, 960, 968, 972, 1029, 1080, 1125, 1152, 1200, 1215, 1250, 1280, 1323, 1350, 1352, 1372, 1408, 1440, 1458

Offset

1,1

Comments

Numbers k > rad(k)^2 that are not perfect powers, where rad = A007947 is the squarefree kernel.

This sequence differs from A389752 since it is missing 144 = 12^2, 324 = 18^2, etc.

Intersection of A303946 and A059172.

Intersection of A303946 and A341645.

Union of A052486 and A366250 = A389752 \ A389864.

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, i.e., m*k not in A001597 (or, equivalently, A131605).

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) \ A001597, j > A010846(k) are the terms in this sequence with rad(s) = k. In this manner, this sequence is R(k,j) \ A001597, 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 is not a term since it is a perfect power and since it is the square of its squarefree kernel.
The number 144 = 12^2 is not a term since it is a perfect power.

Mathematica

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

Crossrefs

Cf. A003557, A007947, A010846, A059404, A059172, A120944, A126706, A303606, A303946, A341645, A366250, A370266, A388304, A389227, A389528, A389864, A389752.

Sequence in context: A080854 A389528 A389752 * A255267 A366250 A345503

Adjacent sequences: A389750 A389751 A389752 * A389754 A389755 A389756

Keyword

nonn

Author

Michael De Vlieger, Nov 11 2025

Status

approved