This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A336025 #18 Jun 15 2024 11:34:02
%S A336025 4,9,20,25,27,28,172,176
%N A336025 Numbers m providing record values for the proportion of nonsquarefree integers in the interval [1, m].
%C A336025 Also, numbers providing record low values for the proportion of squarefree integers.
%C A336025 The proportion of nonsquarefree integers approaches 1-6/Pi^2. For low values of m the proportion in [1, m] tends to be lower, since squares appear late. But values of m for which the ratio in the interval [1, m] is larger than the limit value do exist. The first such one is 28. Therefore this sequence is finite and it can be proved that 176 is indeed its last term. The proportion of nonsquarefree numbers in [1, 176] is 70/176 = 0.397727272... and that of squarefree ones is 0.6022727...
%C A336025 Rogers (1964) proved that if A(x) is the number of squarefree numbers not exceeding x then A(x)/x >= A(176)/176 = 53/88. Therefore 176 is the last term of this sequence. - _Amiram Eldar_, Mar 12 2021
%H A336025 Kenneth Rogers, <a href="https://doi.org/10.1090/S0002-9939-1964-0163893-X">The Schnirelmann density of the squarefree integers</a>, Proceedings of the American Mathematical Society, Vol. 15, No. 4 (1964), pp. 515-516.
%e A336025 Up to m = 9 there are 3 numbers which are divisible by some square: 4, 8 and 9, for a proportion of 3/9 = 1/3. No interval [1, k] for k < 9 has a ratio as high, so 9 is in the sequence.
%t A336025 s = {}; c = 0; rm = 0; Do[If[!SquareFreeQ[n], c++]; If[(r = c/n) > rm, rm = r; AppendTo[s, n]], {n, 1, 10^3}]; s (* _Amiram Eldar_, Mar 12 2021 *)
%t A336025 Module[{nn=200},DeleteDuplicates[{#[[1]],#[[2]]/#[[1]]}&/@Thread[{Range[nn],Accumulate[ Table[If[SquareFreeQ[n],0,1],{n,nn}]]}],GreaterEqual[#1[[2]],#2[[2]]]&]][[;;,1]]//Rest (* _Harvey P. Dale_, Jun 15 2024 *)
%Y A336025 Cf. A005117, A013929, A173143, A057627.
%K A336025 nonn,fini,full
%O A336025 1,1
%A A336025 _Javier Múgica_, Jul 05 2020