OFFSET
1,1
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 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.
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.
FORMULA
{ k : (k-1)*A002819(k) > 0. }
EXAMPLE
906150257 is the smallest number k > 1 with A002819(k) > 0 (see Tanaka 1980).
PROG
(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 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Jonathan Sondow, Jun 13 2011
STATUS
approved