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.

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, 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, 1, 1, 1, 2, 2, 1, 4, 4, 1, 2, 1, 7, 2

Offset

1,2

Comments

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?

Essentially this sequence is a run-length encoding of A008836. - Alonso del Arte, Feb 29 2012

References

H. Gupta, On a table of values of L(n), Proceedings of the Indian Academy of Sciences. Section A, 12 (1940), 407-409.

H. Gupta, A table of values of Liouville's function L(n), Research Bulletin of East Panjab University, No. 3 (Feb. 1950), 45-55.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

R. S. Lehman, On Liouville's function, Math. Comp., 14 (1960), 311-320.

Eric Weisstein's World of Mathematics, Liouville Function

Formula

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).

Example

a(6) = 2 because the 6th Liouville cluster consists of 2 successive negative signs: lambda(7) = lambda(8) = (-1).
a(7) = 2 because the 7th Liouville cluster consists of 2 successive positive signs: lambda(9) = lambda(10) = 1.

Mathematica

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 *)
Length/@Split[LiouvilleLambda[Range[300]]] (* Harvey P. Dale, Jul 02 2017 *)

Prog

(Haskell)
import Data.List (group)
a069545 n = a069545_list !! (n-1)
a069545_list = map length $ group a008836_list
-- Reinhard Zumkeller, Mar 10 2014

Crossrefs

Cf. A008836, A002819, A001222, A028260, A026424.

Sequence in context: A059111 A103502 A127950 * A353524 A122520 A284995

Adjacent sequences: A069542 A069543 A069544 * A069546 A069547 A069548

Keyword

easy,nice,nonn

Author

Paul D. Hanna, Apr 17 2002

Extensions

Corrected a(46) and a(47), and added terms after that. - Alonso del Arte, Feb 29 2012

Status

approved