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A109507
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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).
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1
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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; internal format)
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OFFSET
| 1,3
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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.
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.
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LINKS
| J. J. O'Connor and E. F. Robertson, Atle Selberg.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics..
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FORMULA
| Selberg proved that S(x) = 2*x*ln(x) + o(x*ln(x)).
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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}]
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CROSSREFS
| Cf. A109508.
Sequence in context: A189364 A022797 A190884 * A160802 A170888 A182838
Adjacent sequences: A109504 A109505 A109506 * A109508 A109509 A109510
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KEYWORD
| nonn
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AUTHOR
| Jonathan Vos Post (jvospost3(AT)gmail.com) and Robert G. Wilson v, Jun 30 2005
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