A068088 n-3, n-2, n-1, n+1, n+2 and n+3 are squarefree.

4, 32, 36, 40, 68, 104, 108, 112, 140, 180, 184, 212, 216, 220, 256, 284, 320, 356, 392, 396, 400, 432, 436, 464, 468, 500, 544, 612, 616, 644, 680, 716, 756, 760, 788, 792, 796, 860, 896, 900, 904, 936, 940, 968, 1004, 1008, 1040, 1044, 1112, 1116, 1120, 1156, 1188, 1192, 1220, 1256, 1260, 1264

Offset: 1

Author

Amarnath Murthy, Feb 18 2002

Keywords

• easy
• nonn

Comments

No four consecutive numbers can all be squarefree, as one of them is divisible by 2^2 = 4.

From 28 to 44 there are 12 squarefree numbers among 15 consecutive integers. Other examples are 100 to 116 and 212 to 228.

The largest possible run of consecutive multiples of 4 in the sequence is 3: If n, n+4 and n+8 are in the sequence then n+4 and hence n-5 and n+13 must be divisible by 9, so neither n-4 nor n+12 can be in the sequence. - Ulrich Schimke, Apr 13 2002

Examples

			36 is a term as 33,34,35 and 37,38,39 are two sets of three consecutive squarefree numbers.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007675, A039833. Equals 4*A283628.

Programs

Maple

select(t -> andmap(numtheory:-issqrfree, [t-3,t-2,t-1,t+1,t+2,t+3]), [seq(i,i=4..2000,4)]); # Robert Israel, Jun 05 2018

Mathematica

<< NumberTheory`NumberTheoryFunctions` lst={};Do[If[SquareFreeQ[n-1]&&SquareFreeQ[n+1]&&SquareFreeQ[n-2]&&SquareFreeQ[n+2]&&SquareFreeQ[n-3]&&SquareFreeQ[n+3],AppendTo[lst,n]],{n,7!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 26 2009 *)

Extensions

Corrected and extended by Ulrich Schimke, Apr 13 2002

Further correction from Harvey P. Dale, May 01 2002

Offset changed to 1 by Michel Marcus, May 24 2014