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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).

%I A175201 #34 Mar 04 2023 15:04:22
%S A175201 1,9,14,33,54,140,140,213,213,1934,1934,1934,35811,38405,38405,200938,
%T A175201 200938,389409,1792209,5606457,8405437,8405437,8405437,8405437,
%U A175201 68780189,68780189,68780189,68780189,880346227,880346227,880346227,880346227,880346227
%N 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).
%C A175201 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.
%D A175201 H. Gupta, On a table of values of L(n), Proceedings of the Indian Academy of Sciences. Section A, 12 (1940), 407-409.
%D A175201 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 A175201 Donovan Johnson and Giovanni Resta, <a href="/A175201/b175201.txt">Table of n, a(n) for n = 1..44</a> (terms < 10^13, first 37 terms from Donovan Johnson)
%H A175201 Peter Borwein, Ron Ferguson, and Michael J. Mossinghoff, <a href="http://dx.doi.org/10.1090/S0025-5718-08-02036-X">Sign changes in sums of the Liouville function</a>, Math. Comp. 77 (2008), 1681-1694.
%H A175201 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.
%F A175201 lambda(n) = (-1)^omega(n) where omega(n) is the number of prime factors of n with multiplicity.
%e A175201 a(1) = 1 and L(1) = 1;
%e A175201 a(2) = 9 and L(9) = L(10)= 1;
%e A175201 a(3) = 14 and L(14) = L(15) = L(16) = 1;
%e A175201 a(4) = 33 and L(33) = L(34) = L(35) = L(36) = 1.
%p A175201 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:
%t A175201 Table[k=1;While[Sum[LiouvilleLambda[k+i],{i,0,n-1}]!=n,k++];k,{n,1,30}]
%t A175201 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 *)
%Y A175201 Cf. A172354, A051470, A028488, A002819, A175202, A066793.
%K A175201 nonn
%O A175201 1,2
%A A175201 _Michel Lagneau_, Mar 04 2010