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A175201 a(n) is the smallest k such that the n consecutive values L(k), L(k+1),…, L(k+n-1) = 1, where L(m) is the Liouville function A008836(m). 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Short history of conjecture L(n) <= 0 for all n >= 2 by Deborah Tepper Haimo. 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}]

CROSSREFS

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

Sequence in context: A093021 A272781 A229543 * A066793 A275508 A139055

Adjacent sequences:  A175198 A175199 A175200 * A175202 A175203 A175204

KEYWORD

nonn

AUTHOR

Michel Lagneau, Mar 04 2010

STATUS

approved

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Last modified September 28 17:43 EDT 2020. Contains 337393 sequences. (Running on oeis4.)