A175201 a(n) is the smallest k such that the n consecutive values lambda(k), lambda(k+1), ..., lambda(k+n-1) = 1, where lambda(m) is the Liouville function A008836(m).

1, 9, 14, 33, 54, 140, 140, 213, 213, 1934, 1934, 1934, 35811, 38405, 38405, 200938, 200938, 389409, 1792209, 5606457, 8405437, 8405437, 8405437, 8405437, 68780189, 68780189, 68780189, 68780189, 880346227, 880346227, 880346227, 880346227, 880346227

Offset

1,2

Comments

Short history of conjecture L(n) <= 0 for all n >= 2 by Deborah Tepper Haimo, where L(n) is the summatory Liouville function A002819(n). George Polya conjectured 1919 that L(n) <= 0 for all n >= 2. The conjecture was generally deemed true for nearly 40 years, until 1958, when C. B. Haselgrove proved that L(n) > 0 for infinitely many n. In 1962, R. S. Lehman found that L(906180359) = 1 and in 1980, M. Tanaka discovered that the smallest counterexample of the Polya conjecture occurs when n = 906150257.

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

Donovan Johnson and Giovanni Resta, Table of n, a(n) for n = 1..44 (terms < 10^13, first 37 terms from Donovan Johnson)

Peter Borwein, Ron Ferguson, and Michael J. Mossinghoff, Sign changes in sums of the Liouville function, Math. Comp. 77 (2008), 1681-1694.

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

Formula

lambda(n) = (-1)^omega(n) where omega(n) is the number of prime factors of n with multiplicity.

Example

a(1) = 1 and L(1) = 1;
a(2) = 9 and L(9) = L(10)= 1;
a(3) = 14 and L(14) = L(15) = L(16) = 1;
a(4) = 33 and L(33) = L(34) = L(35) = L(36) = 1.

Maple

with(numtheory):for k from 0 to 30 do : indic:=0:for n from 1 to 1000000000 while (indic=0)do :s:=0:for i from 0 to k do :if (-1)^bigomega(n+i)= 1 then s:=s+1: else fi:od:if s=k+1 and indic=0 then print(n):indic:=1:else fi:od:od:

Mathematica

Table[k=1; While[Sum[LiouvilleLambda[k+i], {i, 0, n-1}]!=n, k++]; k, {n, 1, 30}]
With[{c=LiouvilleLambda[Range[841*10^4]]}, Table[SequencePosition[c, PadRight[ {}, n, 1], 1][[All, 1]], {n, 24}]//Flatten] (* The program generates the first 24 terms of the sequence. *) (* Harvey P. Dale, Jul 27 2022 *)

Crossrefs

Cf. A172354, A051470, A028488, A002819, A175202, A066793.

Sequence in context: A360406 A272781 A229543 * A066793 A275508 A139055

Adjacent sequences: A175198 A175199 A175200 * A175202 A175203 A175204

Keyword

nonn

Author

Michel Lagneau, Mar 04 2010

Status

approved