A233409 Squares with squarefree neighbors.

4, 16, 36, 144, 196, 256, 400, 484, 900, 1156, 1296, 1600, 1764, 2704, 2916, 3136, 3364, 3600, 4356, 5184, 6084, 7056, 7396, 7744, 8100, 8464, 8836, 9216, 10404, 10816, 11236, 11664, 12100, 12544, 12996, 16384, 16900, 19044, 19600, 20164, 20736, 22500

Offset: 1

Author

Irina Gerasimova, Dec 09 2013

Keywords

• nonn

Comments

All terms are multiples of 4. Whether n is congruent to 1 or 3 mod 4, n^2 is congruent to 1 mod 3 and therefore mu(n^2 - 1) = 0. - Alonso del Arte, Dec 12 2013

Examples

			36 is in this sequence because 35 and 37 are both squarefree.
64 is not in this sequence because 63 = 3^2 * 7.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000290, A005117, A088068.

Programs

Mathematica

Select[Table[n^2, {n, 150}], SquareFreeQ[# - 1] && SquareFreeQ[# + 1] &] (* Vaclav Kotesovec, Dec 11 2013 *)
Select[Range[150]^2, Abs[MoebiusMu[# - 1] MoebiusMu[# + 1]] == 1 &] (* Alonso del Arte, Dec 11 2013 *)
SequencePosition[Table[Which[IntegerQ[Sqrt[n]],1,SquareFreeQ[n],2,True,0],{n,25000}],{2,1,2}][[;;,1]]+1 (* Harvey P. Dale, Jun 27 2023 *)

PARI

forstep(n=2,1e3,[2, 2, 6, 2, 2, 2, 2],if(issquarefree(n-1) && issquarefree(n+1) && issquarefree(n^2+1), print1(n^2", "))) \\ Charles R Greathouse IV, Mar 18 2014