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%I #15 Jun 28 2017 13:00:28
%S 0,1,3,7,11,15,20,25,31,35,43,46,55,60,66,71,81,85,95,100,106,112,124,
%T 127,137,143,151,156,169,171,185,192,199,205,214,217,232,238,246,250,
%U 266,268,284,290,296,302,319,323,336,340,349,354,372,376,386,390,399
%N Let x be a positive number, Lambda(d) = Moebius(d)*[log(x/d)]^2, f(m) = Sum_{d|m} Lambda(d), S(x) = Sum_{m <= x} f(m). Sequence gives nearest integer to S(n).
%D T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976.
%D T. Nagell, "Introduction to Number Theory", Chelsea Pub., New York, 1964, Chap. VIII.
%H J. J. O'Connor and E. F. Robertson, <a href="http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Selberg.html">Atle Selberg</a>.
%H Atle Selberg, <a href="http://dx.doi.org/10.4153/CJM-1950-007-5">An elementary proof of the prime number theorem for arithmetic progressions</a>, Canad. J. Math., 2, (1949), 66-78.
%H Atle Selberg, <a href="http://www.jstor.org/stable/1969454">An elementary proof of Dirichlet's theorem about primes in an arithmetic progression</a>, Annals Math., 50, (1949). 297-304.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SelbergsFormula.html">Selberg's Formula</a>.
%F Selberg proved that S(x) = 2*x*log(x) + o(x*log(x)).
%t lmbd[d_, x_] := MoebiusMu[d]*Log[x/d]^2; f[n_, x_] := Block[{d = Divisors[n]}, Plus @@ lmbd[d, x]]; s[x_] := Sum[f[n, x], {n, x}]; Table[ Floor[ s[n]], {n, 57}]
%Y Cf. A109508.
%K nonn
%O 1,3
%A _Jonathan Vos Post_ and _Robert G. Wilson v_, Jun 30 2005