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A189229
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Counterexamples to Polya's conjecture that A002819(n) <= 0 if n > 1.
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3
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906150257, 906150258, 906150259, 906150260, 906150261, 906150262, 906150263, 906150264, 906150265, 906150266, 906150267, 906150268, 906150269, 906150270, 906150271, 906150272, 906150273, 906150274, 906150275, 906150276, 906150277, 906150278, 906150279, 906150280
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OFFSET
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1,1
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COMMENTS
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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.
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REFERENCES
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Barry Mazur and William Stein, Prime Numbers and the Riemann Hypothesis, Cambridge University Press, 2016. See p. 22.
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LINKS
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FORMULA
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EXAMPLE
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906150257 is the smallest number k > 1 with A002819(k) > 0 (see Tanaka 1980).
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PROG
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(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 */
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CROSSREFS
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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).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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