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A069545 Liouville clusters: the number of successive occurrences of signs in Liouville function lambda(k); a(2n-1) is number of successive positive signs, while a(2n) is number of successive negative signs.

%I A069545 #22 Feb 16 2025 08:32:46
%S A069545 1,2,1,1,1,2,2,3,3,4,2,1,3,6,4,1,3,5,1,2,1,1,1,2,5,1,1,1,1,1,2,3,1,4,
%T A069545 1,2,1,3,2,1,5,1,2,1,4,3,1,3,1,1,1,4,1,3,1,2,2,1,3,2,1,2,1,2,5,3,7,3,
%U A069545 1,1,1,2,2,1,4,4,1,2,1,7,2
%N A069545 Liouville clusters: the number of successive occurrences of signs in Liouville function lambda(k); a(2n-1) is number of successive positive signs, while a(2n) is number of successive negative signs.
%C A069545 Related open questions. What is the limit of ratio: a(n)/n, as n->infinity? What is frequency distribution of integer k in the sequence; a(n)=k for what set of n?
%C A069545 Essentially this sequence is a run-length encoding of A008836. - _Alonso del Arte_, Feb 29 2012
%D A069545 H. Gupta, On a table of values of L(n), Proceedings of the Indian Academy of Sciences. Section A, 12 (1940), 407-409.
%D A069545 H. Gupta, A table of values of Liouville's function L(n), Research Bulletin of East Panjab University, No. 3 (Feb. 1950), 45-55.
%H A069545 Reinhard Zumkeller, <a href="/A069545/b069545.txt">Table of n, a(n) for n = 1..10000</a>
%H A069545 R. S. Lehman, <a href="http://dx.doi.org/10.1090/S0025-5718-1960-0120198-5">On Liouville's function</a>, Math. Comp., 14 (1960), 311-320.
%H A069545 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LiouvilleFunction.html">Liouville Function</a>
%F A069545 Related to summatory Liouville function (A002819): L(m)=sum_{k=1, n} (-1)^(k-1)*a(k) where m=sum_{k=1, n} a(k).
%e A069545 a(6) = 2 because the 6th Liouville cluster consists of 2 successive negative signs: lambda(7) = lambda(8) = (-1).
%e A069545 a(7) = 2 because the 7th Liouville cluster consists of 2 successive positive signs: lambda(9) = lambda(10) = 1.
%t A069545 max = 227; lambdaClLens = {}; Module[{curr = 1, cl = 1, iter = 2}, While[iter < max, If[LiouvilleLambda[iter] == curr, cl++, AppendTo[lambdaClLens, cl]; curr = (-1)curr; cl = 1]; iter++]]; lambdaClLens (* _Alonso del Arte_, Feb 29 2012 *)
%t A069545 Length/@Split[LiouvilleLambda[Range[300]]] (* _Harvey P. Dale_, Jul 02 2017 *)
%o A069545 (Haskell)
%o A069545 import Data.List (group)
%o A069545 a069545 n = a069545_list !! (n-1)
%o A069545 a069545_list = map length $ group a008836_list
%o A069545 -- _Reinhard Zumkeller_, Mar 10 2014
%Y A069545 Cf. A008836, A002819, A001222, A028260, A026424.
%K A069545 easy,nice,nonn
%O A069545 1,2
%A A069545 _Paul D. Hanna_, Apr 17 2002
%E A069545 Corrected a(46) and a(47), and added terms after that. - _Alonso del Arte_, Feb 29 2012