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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. 1
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 (list; graph; refs; listen; history; text; internal format)
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 * A122520 A284995 A243005

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

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Last modified October 18 08:08 EDT 2019. Contains 328146 sequences. (Running on oeis4.)