A280892 Squareful numbers with both neighbors squarefree.

4, 12, 16, 18, 20, 32, 36, 40, 52, 54, 56, 60, 68, 72, 84, 88, 90, 92, 96, 104, 108, 112, 128, 132, 140, 144, 150, 156, 160, 162, 164, 180, 184, 192, 196, 198, 200, 204, 212, 216, 220, 228, 232, 234, 236, 240, 248, 250, 252, 256, 264, 268, 270, 272, 284, 292, 294, 300, 304, 306, 308, 312

Offset: 1

Author

Juri-Stepan Gerasimov, Jan 16 2017

Keywords

• nonn
• easy

Comments

Subsequence of A013929 and A067874.

The asymptotic density of this sequence is Product_{p prime} (1 - 2/p^2) - Product_{p prime} (1 - 3/p^2) = 0.197147118033435... (Mossinghoff et al., 2021). - Amiram Eldar, Nov 11 2021, Mar 21 2024

Examples

			4 is in this sequence because 4 = 2^2 is nonsquarefree and both 4 - 1 = 3 and 4 + 1 = 5 are squarefree.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
• Michael J. Mossinghoff, Tomás Oliveira e Silva, and Tim Trudgian, The distribution of k-free numbers, Mathematics of Computation, Vol. 90, No. 328 (2021), pp. 907-929; arXiv preprint, arXiv:1912.04972 [math.NT], 2019-2020.

Crossrefs

Cf. A005117, A013929, A067874, A228649, A272799, A281192.

Cf. A065474, A206256.

Programs

Magma

[n: n in [2..300] | not IsSquarefree(n) and IsSquarefree(n-1) and IsSquarefree(n+1)];

Mathematica

Mean/@SequencePosition[Table[If[SquareFreeQ[n],1,0],{n,400}],{1,0,1}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 19 2020 *)

PARI

isok(n) = !issquarefree(n) && issquarefree(n-1) && issquarefree(n+1); \\ Michel Marcus, Jun 18 2017

Extensions

Definition corrected by Jon E. Schoenfield, Jun 18 2017