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 A109507 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). 1
 0, 1, 3, 7, 11, 15, 20, 25, 31, 35, 43, 46, 55, 60, 66, 71, 81, 85, 95, 100, 106, 112, 124, 127, 137, 143, 151, 156, 169, 171, 185, 192, 199, 205, 214, 217, 232, 238, 246, 250, 266, 268, 284, 290, 296, 302, 319, 323, 336, 340, 349, 354, 372, 376, 386, 390, 399 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 REFERENCES T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976. T. Nagell, "Introduction to Number Theory", Chelsea Pub., New York, 1964, Chap. VIII. LINKS J. J. O'Connor and E. F. Robertson, Atle Selberg. Atle Selberg, An elementary proof of the prime number theorem for arithmetic progressions, Canad. J. Math., 2, (1949), 66-78. Atle Selberg, An elementary proof of Dirichlet's theorem about primes in an arithmetic progression, Annals Math., 50, (1949). 297-304. Eric Weisstein's World of Mathematics, Selberg's Formula. FORMULA Selberg proved that S(x) = 2*x*log(x) + o(x*log(x)). MATHEMATICA 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}] CROSSREFS Cf. A109508. Sequence in context: A310211 A310212 A310213 * A160802 A170888 A182838 Adjacent sequences:  A109504 A109505 A109506 * A109508 A109509 A109510 KEYWORD nonn AUTHOR Jonathan Vos Post and Robert G. Wilson v, Jun 30 2005 STATUS approved

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Last modified October 19 15:50 EDT 2019. Contains 328223 sequences. (Running on oeis4.)