A249482 Numbers n such that the summatory Liouville function L(n) (A002819) is zero and L(n-1)*L(n+1) = -1.

2, 906150256, 906150308, 906150310, 906151576, 906154582, 906154586, 906154604, 906154606, 906154608, 906154758, 906154762, 906154764, 906154768, 906154770, 906154788, 906154794, 906154824, 906154826, 906154828, 906154830, 906154836, 906154838, 906154856

Offset: 1

Author

Vladimir Shevelev, Jan 13 2015

Keywords

• nonn

Comments

To create the data, the author studied the b-file of Donovan Johnson in A189229.

For k>=1,

in the interval [a(2k-1), a(2k)], L(n)<=0,

in the interval [a(2k), a(2k+1)], L(n)>=0.

In particular, for k=1, in the interval [2, 906150256], L(n)<=0.

G. Polya (1919) conjectured that L(n)<=0, for n>=2. But this was disproved in 1958 by B. Haselgrove, and in 1980 M. Tanaka found the smallest counterexample, a(2)+1 = 906150257.

Links

• P. Borwein, R. Ferguson, and M. Mossinghoff, Sign changes in sums of the Liouville function, Mathematics of Computation 77 (2008), pp. 1681-1694.
• M. Tanaka, A Numerical Investigation on Cumulative Sum of the Liouville Function, Tokyo J. Math. 3, 187-189, 1980.
• Eric Weisstein's World of Mathematics, Liouville Function
• Eric Weisstein's World of Mathematics, Polya Conjecture

Crossrefs

Cf. A002819, A028488, A051470, A189229, A249487, A253174.