%I A023900
%S A023900 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,
%T A023900 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,
%U A023900 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
%V A023900 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,
%W A023900 -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,
%X A023900 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
%N A023900 Dirichlet inverse of Euler totient function (A000010).
%C A023900 Also called reciprocity balance of n.
%C A023900 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
%C A023900 Signed: (1, -1, -2, -1, -4, 2, -6,...) = row sums of triangle A143256 
               [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 02 2008]
%D A023900 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 
               1976, page 37.
%D A023900 D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc. Boston, 
               MA, 1976, p. 125.
%H A023900 T. D. Noe, <a href="b023900.txt">Table of n, a(n) for n=1..1000</a>
%F A023900 a(n) = Sum_{ d divides n } d*mu(d) = Product_{p|n} (1-p).
%F A023900 a(n) = 1 / (sum_{ d divides n } mu(d)*d/phi(d)).
%F A023900 Dirichlet g.f.: zeta(s)/zeta(s-1).
%F A023900 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
%F A023900 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
%F A023900 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
%F A023900 G.f.: A(x) is x times the logarithmic derivative of A117209(x). - Stuart 
               Clary (clary(AT)uakron.edu), Apr 15, 2006
%F A023900 Row sums of triangle A134842 - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Nov 12 2007
%F A023900 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]
%t A023900 Array[ Function[ n, 1/Plus @@ Map[ #*MoebiusMu[ # ]/EulerPhi[ # ]&, Divisors[ 
               n ] ] ], 90 ]
%t A023900 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
%o A023900 (PARI) a(n)=direuler(p=2,n,(1-p*X)/(1-X))[n]
%o A023900 (PARI) j=[]; for(n=1,250,j=concat(j,sumdiv(n,d,d*moebius(d)))); j
%Y A023900 Cf. A000010, A023898. Moebius transform is A055615.
%Y A023900 Cf. A117209.
%Y A023900 Cf. A134842.
%Y A023900 Sequence in context: A147763 A098371 A070777 this_sequence A141564 A046791 
               A125131
%Y A023900 Adjacent sequences: A023897 A023898 A023899 this_sequence A023901 A023902 
               A023903
%K A023900 sign,easy,nice,mult
%O A023900 1,3
%A A023900 Olivier Gerard (olivier.gerard(AT)gmail.com)
%E A023900 Additional comments from Michael Somos, Jun 04 2000.