A072284 - OEIS

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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

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OFFSET

1,2

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

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.

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)

EXAMPLE

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.

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 *)

PROG

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

CROSSREFS

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

Sequence in context: A313385 A277269 A185450 * A242898 A230486 A024507

Adjacent sequences: A072281 A072282 A072283 * A072285 A072286 A072287

KEYWORD

easy,nonn

AUTHOR

Joseph L. Pe, Jul 10 2002

EXTENSIONS

More terms from Ralf Stephan, Mar 19 2003

STATUS

approved