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

%I A072284 #24 Feb 16 2025 08:32:46
%S A072284 1,5,10,13,17,19,21,26,29,33,37,41,46,51,53,55,57,61,65,69,73,77,82,
%T A072284 85,89,91,93,97,101,105,109,113,118,122,127,129,133,137,141,145,149,
%U A072284 151,154,157,161,163,165,170,173,177,181,185,190,193,197,199,201,205,209
%N A072284 Numbers k begins a new chain of squarefree integers. I.e., k is squarefree but k-1 is not.
%C A072284 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
%H A072284 Reinhard Zumkeller, <a href="/A072284/b072284.txt">Table of n, a(n) for n = 1..1000</a>
%H A072284 Kaisa Matomäki, Maksym Radziwiłł and Terence Tao, <a href="https://doi.org/10.1017/fms.2016.6">Sign patterns of the Liouville and Möbius functions</a>, Forum of Mathematics, Sigma, Vol. 4. (2016), e14.
%H A072284 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Squarefree.html">Squarefree</a>.
%F A072284 From _Reinhard Zumkeller_, Jan 20 2008: (Start)
%F A072284 A136742 mod a(n) = 0;
%F A072284 A136742(n) = Product_{k=0..A120992(n)-1} (a(n) + k);
%F A072284 A136743(n) = Sum_{k=0..A120992(n)-1} A001221(a(n) + k). (End)
%e A072284 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.
%t A072284 Select[Range[100], MoebiusMu[# - 1] == 0  && Abs[MoebiusMu[#]] == 1 &] (* _Amiram Eldar_, Feb 14 2021 *)
%t A072284 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 *)
%o A072284 (PARI) n=1; for(k=1,100, while(!issquarefree(n),n=n+1); print1(n","); while(issquarefree(n),n=n+1))
%Y A072284 Cf. A001221, A005117, A013929, A120992, A136742, A136743.
%K A072284 easy,nonn
%O A072284 1,2
%A A072284 _Joseph L. Pe_, Jul 10 2002
%E A072284 More terms from _Ralf Stephan_, Mar 19 2003