cp's OEIS Frontend

A068781 Lesser of two consecutive numbers each divisible by a square.

%I A068781 #32 Sep 17 2024 22:44:00
%S A068781 8,24,27,44,48,49,63,75,80,98,99,116,120,124,125,135,147,152,168,171,
%T A068781 175,188,207,224,242,243,244,260,275,279,288,296,315,324,332,342,343,
%U A068781 350,351,360,363,368,375,387,404,423,424,440,459,475,476,495,507,512
%N A068781 Lesser of two consecutive numbers each divisible by a square.
%C A068781 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 A068781 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 A068781 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 A068781 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
%C A068781 Maximum of the n-th maximal anti-run of nonsquarefree numbers (A013929) differing by more than one. For runs instead of anti-runs we have A376164. For squarefree instead of nonsquarefree we have A007674. - _Gus Wiseman_, Sep 14 2024
%H A068781 Reinhard Zumkeller, <a href="/A068781/b068781.txt">Table of n, a(n) for n = 1..10000</a>
%H A068781 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 A068781 A261869(a(n)) = 0. - _Reinhard Zumkeller_, Sep 04 2015
%e A068781 44 is in the sequence because 44 = 2^2 * 11 and 45 = 3^2 * 5.
%e A068781 From _Gus Wiseman_, Sep 14 2024: (Start)
%e A068781 Splitting nonsquarefree numbers into maximal anti-runs gives:
%e A068781   (4,8)
%e A068781   (9,12,16,18,20,24)
%e A068781   (25,27)
%e A068781   (28,32,36,40,44)
%e A068781   (45,48)
%e A068781   (49)
%e A068781   (50,52,54,56,60,63)
%e A068781   (64,68,72,75)
%e A068781   (76,80)
%e A068781   (81,84,88,90,92,96,98)
%e A068781   (99)
%e A068781 The maxima are a(n). The corresponding pairs are (8,9), (24,25), (27,28), (44,45), etc.
%e A068781 (End)
%t A068781 Select[ Range[2, 600], Max[ Transpose[ FactorInteger[ # ]] [[2]]] > 1 && Max[ Transpose[ FactorInteger[ # + 1]] [[2]]] > 1 &]
%t A068781 f@n_:= Flatten@Position[Partition[SquareFreeQ/@Range@2000,n,1], Table[False,{n}]]; f@2 (* _Hans Rudolf Widmer_, Aug 30 2022 *)
%t A068781 Max/@Split[Select[Range[100], !SquareFreeQ[#]&],#1+1!=#2&]//Most (* _Gus Wiseman_, Sep 14 2024 *)
%o A068781 (Haskell)
%o A068781 a068781 n = a068781_list !! (n-1)
%o A068781 a068781_list = filter ((== 0) . a261869) [1..]
%o A068781 -- _Reinhard Zumkeller_, Sep 04 2015
%o A068781 (PARI) isok(m) = !moebius(m) && !moebius(m+1); \\ _Michel Marcus_, Feb 14 2021
%Y A068781 Cf. A068780, A068140, A068781, A068782, A068783, A068784, A068785.
%Y A068781 Cf. A049535, A077647, A078143, A045882.
%Y A068781 Subsequence of A261869.
%Y A068781 Cf. A007674, A373409, A373410, A373412, A375707, A376164.
%Y A068781 A005117 lists the squarefree numbers, first differences A076259.
%Y A068781 A013929 lists the nonsquarefree numbers, first differences A078147.
%Y A068781 A053797 gives lengths of runs of nonsquarefree numbers, firsts A373199.
%Y A068781 Cf. A049094, A061399, A072284, A120992, A294242, A373573, A375709.
%K A068781 nonn
%O A068781 1,1
%A A068781 _Robert G. Wilson v_, Mar 04 2002