A390538 - OEIS

%I #7 Nov 10 2025 09:15:54

%S 1,2,3,4,5,6,7,9,10,11,12,13,14,15,17,18,19,20,21,22,23,24,25,26,28,

%T 29,30,31,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,49,50,51,52,53,

%U 55,56,57,58,59,60,61,62,63,65,66,67,68,69,70,71,73,74,75,76

%N Numbers that do not exceed the square of their squarefree kernel.

%C Numbers k such that k <= rad(k)^2, where rad = A007947.

%C Complement of A059172.

%C Union of {1}, A177492, and A341646.

%C Define sequence S(r) to be the set {m*r : rad(m) | r, m >= 1} for squarefree r (i.e., r in A005117). 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 S(r,j) is the j-th term in S(r), that this sequence contains S(r,j) for j = 1..A010846(r). Therefore this sequence is a superset of A120944.

%C For prime p, the intersection of this sequence and S(p) = {p^m : m >= 1} is {p, p^2}. Therefore, this sequence is a superset of primes (A000040) and prime squares (A001248).

%H Michael De Vlieger, <a href="/A390538/b390538.txt">Table of n, a(n) for n = 1..10000</a>

%e For r = 1, S(1) = {1}, 1 does not exceed 1^2, therefore a(1) = 1.

%e For r = 2, this sequence contains S(r,j) = A000079(j), j = 1..A010846(2), i.e., {2, 4}.

%e For r = 6, this sequence contains S(r,j) = A033485(j), j = 1..A010846(6), i.e., {6, 12, 18, 24, 36}.

%e For r = 10, this sequence contains S(r,j) = A033486(j), j = 1..A010846(10), i.e., {10, 20, 40, 50, 80, 100}. Therefore, a(7) = 160 is the first term in this sequence that is not in A033845.

%t Select[Range[76], # <= Apply[Times, FactorInteger[#][[All, 1]]^2] &]

%Y Cf. A000040, A001248, A003557, A005117, A007947, A059172, A120944, A177492, A341646.

%K nonn

%O 1,2

%A _Michael De Vlieger_, Nov 09 2025