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

%I A189229 #35 Mar 22 2024 19:38:04
%S A189229 906150257,906150258,906150259,906150260,906150261,906150262,
%T A189229 906150263,906150264,906150265,906150266,906150267,906150268,
%U A189229 906150269,906150270,906150271,906150272,906150273,906150274,906150275,906150276,906150277,906150278,906150279,906150280
%N A189229 Counterexamples to Polya's conjecture that A002819(n) <= 0 if n > 1.
%C A189229 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
%C A189229 906150294 is the smallest number > 906150257 that is not in the sequence (see A028488).
%C A189229 See A002819, A008836, A028488, A051470 for additional comments, references, and links.
%C A189229 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.
%D A189229 Barry Mazur and William Stein, Prime Numbers and the Riemann Hypothesis, Cambridge University Press, 2016. See p. 22.
%H A189229 Donovan Johnson, <a href="/A189229/b189229.txt">Table of n, a(n) for n = 1..10000</a>
%H A189229 R. P. Brent and J. van de Lune, <a href="http://arxiv.org/abs/1112.4911">A note on Polya's observation concerning Liouville's function</a>, arXiv:1112.4911 [math.NT] 2011.
%H A189229 Jarosław Grytczuk, <a href="https://arxiv.org/abs/2003.02887">From the 1-2-3 Conjecture to the Riemann Hypothesis</a>, arXiv:2003.02887 [math.CO], 2020. See p. 9.
%H A189229 Ben Sparks, <a href="https://www.youtube.com/watch?v=eQCUPQdi6DY">906,150,257 and the Pólya conjecture (MegaFavNumbers)</a>, SparksMath video (2020).
%H A189229 M. Tanaka, <a href="https://doi.org/10.3836/tjm/1270216093">A Numerical Investigation on Cumulative Sum of the Liouville Function</a>, Tokyo J. Math. 3 (1980), 187-189.
%H A189229 Wikipedia, <a href="http://en.wikipedia.org/wiki/Pólya_conjecture">Pólya conjecture</a>.
%F A189229 { k : (k-1)*A002819(k) > 0. }
%e A189229 906150257 is the smallest number k > 1 with A002819(k) > 0 (see Tanaka 1980).
%o A189229 (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 */
%Y A189229 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).
%K A189229 nonn
%O A189229 1,1
%A A189229 _Jonathan Sondow_, Jun 13 2011