A051470 a(n) is least value of m for which the sum of Liouville's function from 1 to m is n.

1, 906150258, 906150259, 906150260, 906150263, 906150264, 906150331, 906150334, 906150337, 906150338, 906150339, 906150358, 906150359, 906150362, 906150363, 906150368, 906150387, 906150388, 906150389, 906150406, 906150407

Offset: 1

Author

Jud McCranie

Keywords

• nonn

Comments

It was once conjectured that the sum of Liouville's function was never > 0 except for the first term.

It follows from Theorem 2 in Borwein-Ferguson-Mossinghoff that a(n) < 262*n^2 infinitely often, improving on an earlier result of Anderson & Stark. - Charles R Greathouse IV, Jun 14 2011

a(830) > 2 * 10^14 (probably around 3.511e14) and a(1160327) = 351753358289465 according to the calculations of Borwein, Ferguson, & Mossinghoff. - Charles R Greathouse IV, Jun 14 2011

3.75 * 10^14 < a(1160328) <= 23156359315279877168. - Hiroaki Yamanouchi, Oct 04 2015

From Jianing Song, Aug 06 2021: (Start)

a(n) is the smallest m such that A002819(m) = n.

This sequence is strictly increasing since A002819(m) - A002819(m-1) = A008836(m) = +-1. (End)

Examples

			The sum of Liouville's function from 1 through 906150258 is 2, that is the smallest value, so a(2)=906150258.
		

References

• R. J. Anderson and H. M. Stark, Oscillation theorems, Analytic Number Theory (1980); Lecture Notes in Mathematics 899 (1981), pp. 79-106.

Links

• Donovan Johnson and Hiroaki Yamanouchi, Table of n, a(n) for n = 1..100000 (terms a(1)-a(829) from _Donovan Johnson_)
• P. Borwein, R. Ferguson, and M. Mossinghoff, Sign changes in sums of the Liouville function, Mathematics of Computation 77 (2008), pp. 1681-1694.
• R. S. Lehman, On Liouville's function, Math. Comp., 14 (1960), 311-320.
• M. Tanaka, A Numerical Investigation on Cumulative Sum of the Liouville Function, Tokyo J. Math. 3, 187-189, 1980.

Crossrefs

Cf. A008836 (Liouville's function), A002819, A028488.

Programs

PARI

print1(r=1);t=0;for(n=906150257,906400000,t+=(-1)^bigomega(n);if(t>r,r=t;print1(", "n))) \\ Charles R Greathouse IV, Jun 14 2011