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Search: id:A109507
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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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+0
2
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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; listen)
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OFFSET
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1,3
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REFERENCES
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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
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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
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Selberg proved that S(x) = 2*x*ln(x) + o(x*ln(x)).
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MATHEMATICA
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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
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Cf. A109508.
Sequence in context: A039957 A079422 A022797 this_sequence A160802 A163094 A022800
Adjacent sequences: A109504 A109505 A109506 this_sequence A109508 A109509 A109510
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KEYWORD
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nonn
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AUTHOR
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Jonathan Vos Post (jvospost3(AT)gmail.com) and Robert G. Wilson v, Jun 30 2005
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