A189229 Counterexamples to Polya's conjecture that A002819(n) <= 0 if n > 1.

906150257, 906150258, 906150259, 906150260, 906150261, 906150262, 906150263, 906150264, 906150265, 906150266, 906150267, 906150268, 906150269, 906150270, 906150271, 906150272, 906150273, 906150274, 906150275, 906150276, 906150277, 906150278, 906150279, 906150280

Offset: 1

Author

Jonathan Sondow, Jun 13 2011

Keywords

• nonn

Comments

The point is that for all x < 906150257 there are more n <= x with Omega(n) odd than with Omega(n) even. At x = 906150257 the evens go ahead for the first time. - N. J. A. Sloane, Feb 10 2022

906150294 is the smallest number > 906150257 that is not in the sequence (see A028488).

See A002819, A008836, A028488, A051470 for additional comments, references, and links.

See Brent and van de Lune (2011) for a history of Polya's conjecture and a proof that it is true "on average" in a certain precise sense.

Examples

			906150257 is the smallest number k > 1 with A002819(k) > 0 (see Tanaka 1980).
		

References

• Barry Mazur and William Stein, Prime Numbers and the Riemann Hypothesis, Cambridge University Press, 2016. See p. 22.

Links

• Donovan Johnson, Table of n, a(n) for n = 1..10000
• R. P. Brent and J. van de Lune, A note on Polya's observation concerning Liouville's function, arXiv:1112.4911 [math.NT] 2011.
• Jarosław Grytczuk, From the 1-2-3 Conjecture to the Riemann Hypothesis, arXiv:2003.02887 [math.CO], 2020. See p. 9.
• Ben Sparks, 906,150,257 and the Pólya conjecture (MegaFavNumbers), SparksMath video (2020).
• M. Tanaka, A Numerical Investigation on Cumulative Sum of the Liouville Function, Tokyo J. Math. 3 (1980), 187-189.
• Wikipedia, Pólya conjecture.

Crossrefs

Cf. A002819 (Liouville's summatory function L(n)), A008836 (Liouville's function lambda(n)), A028488 (n such that L(n) = 0), A051470 (least m for which L(m) = n).

Programs

PARI

s=1; c=0; for(n=2, 906188859, s=s+(-1)^bigomega(n); if(s>0, c++; write("b189229.txt", c " " n))) /* Donovan Johnson, Apr 25 2013 */

Formula

{ k : (k-1)*A002819(k) > 0. }