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

1,1

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

Links

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A000290, A005117, A088068.

Sequence in context: A075408 A206981 A318149 * A181795 A277285 A136404

Adjacent sequences: A233406 A233407 A233408 * A233410 A233411 A233412

Keyword

nonn

Author

Irina Gerasimova, Dec 09 2013

Status

approved