A228649 Numbers n such that n-1, n and n+1 are all squarefree.

2, 6, 14, 22, 30, 34, 38, 42, 58, 66, 70, 78, 86, 94, 102, 106, 110, 114, 130, 138, 142, 158, 166, 178, 182, 186, 194, 202, 210, 214, 218, 222, 230, 238, 254, 258, 266, 282, 286, 302, 310, 318, 322, 330, 346, 354, 358, 366, 382, 390, 394, 398, 402, 410, 418, 430, 434, 438, 446, 454, 462, 466, 470, 482, 498

Offset

1,1

Comments

Equivalently, a positive integer n is comfortably squarefree if and only if n^3 - n is squarefree. The 'if' direction is obvious from the factorization n(n-1)(n+1), and the converse follows from the coprimality of n, n - 1 and n + 1.

The asymptotic density of comfortably squarefree numbers is the product over all primes of 1 - 3/p^2, which is A206256 = 0.125486980905....

See also comments in A007675.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Dan Asimov, Interesting sequence on MathOverflow, Math-fun mailing list, Mar 28 2023.

Ewan Delanoy, Are there infinitely many triples of consecutive square-free integers?, Math Overflow.

A. P. Goucher, Comfortably squarefree numbers, Complex Projective 4-Space.

Fredrick M. Nelson, Does a(0)=6, a(n+1)=a(n)^3-a(n), define a square-free sequence?, MathOverflow, Mar 24 2023.

Formula

a(n) = A007675(n) + 1. - Giovanni Resta, Aug 29 2013

Maple

with(numtheory):
a := n -> `if`(issqrfree(n-1) and issqrfree(n) and issqrfree(n+1), n, NULL);
seq(a(n), n = 1..500); # Peter Luschny, Jan 18 2014

Mathematica

Select[Range[500], (SquareFreeQ[# - 1] && SquareFreeQ[#] && SquareFreeQ[# + 1]) &] (* Adam P. Goucher *)
Select[Range[2, 500, 2], (MoebiusMu[# - 1] MoebiusMu[#] MoebiusMu[# + 1]) != 0 &] (* Alonso del Arte, Jan 16 2014 *)
Flatten[Position[Partition[Boole[SquareFreeQ/@Range[500]], 3, 1], {1, 1, 1}]]+1 (* Harvey P. Dale, Jan 14 2015 *)
SequencePosition[Table[If[SquareFreeQ[n], 1, 0], {n, 500}], {1, 1, 1}][[All, 1]]+1 (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 02 2018 *)

Prog

(PARI) is(n)=issquarefree(n-1)&&issquarefree(n)&&issquarefree(n+1) \\ Charles R Greathouse IV, Aug 29 2013

Crossrefs

Sequence in context: A101572 A080766 A262506 * A268641 A162796 A172304

Adjacent sequences: A228646 A228647 A228648 * A228650 A228651 A228652

Keyword

easy,nonn

Author

Adam P. Goucher, Aug 29 2013

Status

approved