A389752 - OEIS

%I #7 Nov 14 2025 04:56:26

%S 48,54,72,96,108,144,160,162,192,200,224,250,288,320,324,375,384,392,

%T 400,405,432,448,486,500,567,576,640,648,675,686,704,768,784,800,832,

%U 864,896,960,968,972,1029,1080,1125,1152,1200,1215,1250,1280,1323,1350,1352

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

%C 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.

%C Intersection of A059404 and A059172.

%C Intersection of A059404 and A341645.

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

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

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

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

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

%C 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.

%C 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.

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

%H Michael De Vlieger, <a href="/A389752/b389752.txt">Table of n, a(n) for n = 1..10000</a>

%e A059404(1) = 12 is not a term since 12 < 36.

%e 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.

%e 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.

%e 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.

%e 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.

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

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

%K nonn

%O 1,1

%A _Michael De Vlieger_, Nov 11 2025