A028488 Numbers k such that the summatory Liouville function L(k) (A002819) is zero.

2, 4, 6, 10, 16, 26, 40, 96, 586, 906150256, 906150294, 906150308, 906150310, 906150314, 906151516, 906151576, 906152172, 906154582, 906154586, 906154590, 906154594, 906154604, 906154606, 906154608, 906154758, 906154760, 906154762

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• nice

Comments

a(253) > 2*10^14 according to the calculations of Borwein, Ferguson, & Mossinghoff. Most likely a(253) = 351100332278250. - Charles R Greathouse IV, Jun 14 2011

L(23156358837978983978) = 0 and L(k) < 0 for k from 2.3156354*10^19 to 23156358837978983977. - Hiroaki Yamanouchi, Oct 03 2015

All terms are even since k and A002819(k) have the same parity. - Jianing Song, Aug 06 2021

According to Pólya, numbers (p-3)/4 are members of this sequence, with p a Heegner number > 7 (that is, p is one of 11, 19, 43, 67, and 163). - Friedjof Tellkamp, Feb 15 2025

Links

• Donovan Johnson and Hiroaki Yamanouchi, Table of n, a(n) for n = 1..317312 (a(1)-a(252) 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.
• G. Pólya, Verschiedene Bemerkungen zur Zahlentheorie, Jahresber. DMV 28, 31-40, 1919.
• M. Tanaka, A Numerical Investigation on Cumulative Sum of the Liouville Function, Tokyo J. Math. 3, 187-189, 1980.
• Hiroaki Yamanouchi, Values of L(n) from 2*10^14 to 3.75*10^14 (interval = 5*10^9)
• Eric Weisstein's World of Mathematics, Liouville Function
• Eric Weisstein's World of Mathematics, Polya Conjecture

Crossrefs

Cf. A008836 (Liouville's function), A002819, A051470.

Cf. A003173 (Heegner numbers).

Programs

Maple

B:= [seq((-1)^numtheory:-bigomega(i),i=1..10^5)]:
L:= ListTools:-PartialSums(B):
select(t -> L[t]=0, [$1..10^5]); # Robert Israel, Aug 27 2015

Mathematica

Position[Table[Sum[LiouvilleLambda@ k, {k, 1, n}], {n, 1000}], n_ /; n == 0] // Flatten (* Michael De Vlieger, Aug 27 2015 *)
Position[Accumulate[LiouvilleLambda[Range[1000]]],0]//Flatten (* Harvey P. Dale, Aug 10 2022 *)

Extensions

More terms from Hans Havermann, Jun 24 2002