A336025 Numbers m providing record values for the proportion of nonsquarefree integers in the interval [1, m].

4, 9, 20, 25, 27, 28, 172, 176

Offset: 1

Author

Javier Múgica, Jul 05 2020

Keywords

• nonn
• fini
• full

Comments

Also, numbers providing record low values for the proportion of squarefree integers.

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...

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

Examples

			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.
		

Links

• Kenneth Rogers, The Schnirelmann density of the squarefree integers, Proceedings of the American Mathematical Society, Vol. 15, No. 4 (1964), pp. 515-516.

Crossrefs

Cf. A005117, A013929, A173143, A057627.

Programs

Mathematica

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 *)
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 *)