A072284 Numbers k begins a new chain of squarefree integers. I.e., k is squarefree but k-1 is not.

1, 5, 10, 13, 17, 19, 21, 26, 29, 33, 37, 41, 46, 51, 53, 55, 57, 61, 65, 69, 73, 77, 82, 85, 89, 91, 93, 97, 101, 105, 109, 113, 118, 122, 127, 129, 133, 137, 141, 145, 149, 151, 154, 157, 161, 163, 165, 170, 173, 177, 181, 185, 190, 193, 197, 199, 201, 205, 209

Offset: 1

Author

Joseph L. Pe, Jul 10 2002

Keywords

• easy
• nonn

Comments

The asymptotic density of this sequence is 1/zeta(2) - Product_{p prime} (1 - 2/p^2) = A059956 - A065474 = 0.2852930029... (Matomäki et al., 2016) - Amiram Eldar, Feb 14 2021

Examples

			1 begins a new chain 1, 2, 3 of squarefree integers. 4 is not squarefree. Then 5 begins a new chain 5, 6, 7 of squarefree integers. Hence 1 and 5 are terms of the sequence.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
• Kaisa Matomäki, Maksym Radziwiłł and Terence Tao, Sign patterns of the Liouville and Möbius functions, Forum of Mathematics, Sigma, Vol. 4. (2016), e14.
• Eric Weisstein's World of Mathematics, Squarefree.

Crossrefs

Cf. A001221, A005117, A013929, A120992, A136742, A136743.

Programs

Mathematica

Select[Range[100], MoebiusMu[# - 1] == 0  && Abs[MoebiusMu[#]] == 1 &] (* Amiram Eldar, Feb 14 2021 *)
SequencePosition[Table[If[SquareFreeQ[n],1,0],{n,0,250}],{0,1}][[All,2]]-1 (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 24 2021 *)

PARI

n=1; for(k=1,100, while(!issquarefree(n),n=n+1); print1(n","); while(issquarefree(n),n=n+1))

Formula

From Reinhard Zumkeller, Jan 20 2008: (Start)

A136742 mod a(n) = 0;

A136742(n) = Product_{k=0..A120992(n)-1} (a(n) + k);

A136743(n) = Sum_{k=0..A120992(n)-1} A001221(a(n) + k). (End)

Extensions

More terms from Ralf Stephan, Mar 19 2003