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

1,1

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

Links

Table of n, a(n) for n=1..8.

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

Example

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.

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 *)

Crossrefs

Cf. A005117, A013929, A173143, A057627.

Sequence in context: A023377 A187408 A211055 * A308482 A136769 A351600

Adjacent sequences: A336022 A336023 A336024 * A336026 A336027 A336028

Keyword

nonn,fini,full

Author

Javier Múgica, Jul 05 2020

Status

approved