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%I #27 Sep 07 2022 10:58:22
%S 8,24,27,44,48,49,63,75,80,98,99,116,120,124,125,135,147,152,168,171,
%T 175,188,207,224,242,243,244,260,275,279,288,296,315,324,332,342,343,
%U 350,351,360,363,368,375,387,404,423,424,440,459,475,476,495,507,512
%N Lesser of two consecutive numbers each divisible by a square.
%C Also numbers m such that mu(m)=mu(m+1)=0, where mu is the Moebius-function (A008683); A081221(a(n))>1. - _Reinhard Zumkeller_, Mar 10 2003
%C The sequence contains an infinite family of arithmetic progressions like {36a+8}={8,44,80,116,152,188,...} ={4(9a+2)}. {36a+9} provides 2nd nonsquarefree terms. Such AP's can be constructed to any term by solution of a system of linear Diophantine equation. - _Labos Elemer_, Nov 25 2002
%C 1. 4k^2 + 4k is a member for all k; i.e., 8 times a triangular number is a member. 2. (4k+1) times an odd square - 1 is a member. 3. (4k+3) times odd square is a member. - _Amarnath Murthy_, Apr 24 2003
%C The asymptotic density of this sequence is 1 - 2/zeta(2) + Product_{p prime} (1 - 2/p^2) = 1 - 2 * A059956 + A065474 = 0.1067798952... (Matomäki et al., 2016). - _Amiram Eldar_, Feb 14 2021
%H Reinhard Zumkeller, <a href="/A068781/b068781.txt">Table of n, a(n) for n = 1..10000</a>
%H 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.
%F A261869(a(n)) = 0. - _Reinhard Zumkeller_, Sep 04 2015
%e 44 is in the sequence because 44 = 2^2 * 11 and 45 = 3^2 * 5.
%t Select[ Range[2, 600], Max[ Transpose[ FactorInteger[ # ]] [[2]]] > 1 && Max[ Transpose[ FactorInteger[ # + 1]] [[2]]] > 1 &]
%t f@n_:= Flatten@Position[Partition[SquareFreeQ/@Range@2000,n,1], Table[False,{n}]]; f@2 (* _Hans Rudolf Widmer_, Aug 30 2022 *)
%o (Haskell)
%o a068781 n = a068781_list !! (n-1)
%o a068781_list = filter ((== 0) . a261869) [1..]
%o -- _Reinhard Zumkeller_, Sep 04 2015
%o (PARI) isok(m) = !moebius(m) && !moebius(m+1); \\ _Michel Marcus_, Feb 14 2021
%Y Cf. A068780, A068140, A068781, A068782, A068783, A068784, A068785.
%Y Cf. A049535, A077647, A078143, A045882.
%Y Subsequence of A261869.
%K nonn
%O 1,1
%A _Robert G. Wilson v_, Mar 04 2002
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