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From Von Sterneck's conjecture: floor(sqrt(n)) - 2*|Mertens's function A002321(n)|.
1

%I #18 Mar 19 2017 00:58:44

%S -1,1,-1,0,-2,0,-2,-2,-1,1,-1,-1,-3,-1,1,2,0,0,-2,-2,0,2,0,0,1,3,3,3,

%T 1,-1,-3,-3,-1,1,3,4,2,4,6,6,4,2,0,0,0,2,0,0,1,1,3,3,1,1,3,3,5,7,5,5,

%U 3,5,5,6,8,6,4,4,6,4,2,2,0,2,2,2,4,2,0,0,1,3,1,1,3,5,7,7,5,5,7,7,9

%N From Von Sterneck's conjecture: floor(sqrt(n)) - 2*|Mertens's function A002321(n)|.

%C Von Sterneck conjectured that 2*|A002321(n)| < sqrt(n) for all sufficiently large n. This is now known to be false. This is different from the Mertens conjecture that |A002321(n)| < sqrt(n) for all n > 1 (which is also false).

%D D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section VI.2, p. 188.

%t Table[Floor[Sqrt[n]] - 2 Abs[Plus @@ MoebiusMu[Range[n]]], {n, 1, 80}] (* _Carl Najafi_, Aug 17 2011 *)

%Y Cf. A002321, A059571.

%K sign

%O 1,5

%A _N. J. A. Sloane_, Feb 16 2001