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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

Table of n, a(n) for n=1..57.

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: A189364 A022797 A190884 * 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 December 11 16:02 EST 2017. Contains 295905 sequences.