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Search: id:A023900
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| 1, -1, -2, -1, -4, 2, -6, -1, -2, 4, -10, 2, -12, 6, 8, -1, -16, 2, -18, 4, 12, 10, -22, 2, -4, 12, -2, 6, -28, -8, -30, -1, 20, 16, 24, 2, -36, 18, 24, 4, -40, -12, -42, 10, 8, 22, -46, 2, -6, 4, 32, 12, -52, 2, 40, 6, 36, 28, -58, -8, -60, 30, 12, -1, 48, -20, -66, 16, 44, -24, -70, 2, -72, 36, 8, 18, 60, -24, -78, 4, -2
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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Also called reciprocity balance of n.
Apart from different signs, same as sum( d divides n,core(d)*mu(n/d)), where core(d) (A007913) is the square-free part of d. - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 06 2002
Signed: (1, -1, -2, -1, -4, 2, -6,...) = row sums of triangle A143256 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 02 2008]
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REFERENCES
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T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 37.
D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc. Boston, MA, 1976, p. 125.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
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FORMULA
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a(n) = Sum_{ d divides n } d*mu(d) = Product_{p|n} (1-p).
a(n) = 1 / (sum_{ d divides n } mu(d)*d/phi(d)).
Dirichlet g.f.: zeta(s)/zeta(s-1).
a(n+1)=det(n+1)/det(n) where det(n) is the determinant of the n X n matrix M_(i, j)=i/gcd(i, j)=lcm(i, j)/j - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 19 2003
a(n) = -phi(n)*moebius(A007947(n))*A007947(n)/n. Logarithmic g.f.: Sum_{n>=1} a(n)*x^n/n = log(F(x)) where F(x) is the g.f. of A117209 and satisfies: 1/(1-x) = product_{n>=1} F(x^n). - Paul D. Hanna (pauldhanna(AT)juno.com), Mar 03 2006
G.f.: A(x) = sum_{k>=1} mu(k) k x^k/(1 - x^k) where mu(k) is the Moebius (Mobius) function, A008683 - Stuart Clary (clary(AT)uakron.edu), Apr 15, 2006
G.f.: A(x) is x times the logarithmic derivative of A117209(x). - Stuart Clary (clary(AT)uakron.edu), Apr 15, 2006
Row sums of triangle A134842 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 12 2007
G.f.: x/(1-x) = Sum_{n>=1} a(n)*x^n/(1-x^n)^2. [From Paul D. Hanna (pauldhanna(AT)juno.com), Aug 16 2008]
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MATHEMATICA
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Array[ Function[ n, 1/Plus @@ Map[ #*MoebiusMu[ # ]/EulerPhi[ # ]&, Divisors[ n ] ] ], 90 ]
nmax = 81; Drop[ CoefficientList[ Series[ Sum[ MoebiusMu[k] k x^k/(1 - x^k), {k, 1, nmax} ], {x, 0, nmax} ], x ], 1 ] - Stuart Clary (clary(AT)uakron.edu), Apr 15, 2006
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PROGRAM
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(PARI) a(n)=direuler(p=2, n, (1-p*X)/(1-X))[n]
(PARI) j=[]; for(n=1, 250, j=concat(j, sumdiv(n, d, d*moebius(d)))); j
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CROSSREFS
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Cf. A000010, A023898. Moebius transform is A055615.
Cf. A117209.
Cf. A134842.
Sequence in context: A147763 A098371 A070777 this_sequence A141564 A046791 A125131
Adjacent sequences: A023897 A023898 A023899 this_sequence A023901 A023902 A023903
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KEYWORD
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sign,easy,nice,mult
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AUTHOR
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Olivier Gerard (olivier.gerard(AT)gmail.com)
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EXTENSIONS
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Additional comments from Michael Somos, Jun 04 2000.
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