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%I #225 Feb 03 2024 10:16:19
%S 1,-1,-2,-1,-4,2,-6,-1,-2,4,-10,2,-12,6,8,-1,-16,2,-18,4,12,10,-22,2,
%T -4,12,-2,6,-28,-8,-30,-1,20,16,24,2,-36,18,24,4,-40,-12,-42,10,8,22,
%U -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
%N Dirichlet inverse of Euler totient function (A000010).
%C Also called reciprocity balance of n.
%C Apart from different signs, same as Sum_{d divides n} core(d)*mu(n/d), where core(d) (A007913) is the squarefree part of d. - _Benoit Cloitre_, Apr 06 2002
%C Main diagonal of A191898. - _Mats Granvik_, Jun 19 2011
%D T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 37.
%D D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc. Boston, MA, 1976, p. 125.
%H Antti Karttunen, <a href="/A023900/b023900.txt">Table of n, a(n) for n = 1..20000</a> (first 1000 terms from T. D. Noe)
%H G. P. Brown, <a href="http://www.jstor.org/stable/3621931">Some comments on inverse arithmetic functions</a>, Math. Gaz. 89 (516) (2005) 403-408.
%H K. Dohmen and M. Trinks, <a href="http://arxiv.org/abs/1404.5480">An Abstraction of Whitney's Broken Circuit Theorem</a>, arXiv preprint arXiv:1404.5480 [math.CO], 2014.
%H R. Kemp, <a href="http://dx.doi.org/10.1016/0012-365X(82)90123-6">On the number of words in the language {w in Sigma* | w = w^R }^2</a>, Discrete Math., 40 (1982), 225-234.
%H Paul W. Oxby, <a href="https://arxiv.org/abs/2011.10546">A Function Based on Chebyshev Polynomials as an Alternative to the Sinc Function in FIR Filter Design</a>, arXiv:2011.10546 [eess.SP], 2020.
%H László Tóth, <a href="http://arxiv.org/abs/1310.7053">Multiplicative arithmetic functions of several variables: a survey</a>, arXiv preprint arXiv:1310.7053 [math.NT], 2013.
%F a(n) = Sum_{ d divides n } d*mu(d) = Product_{p|n} (1-p).
%F a(n) = 1 / (Sum_{ d divides n } mu(d)*d/phi(d)).
%F Dirichlet g.f.: zeta(s)/zeta(s-1). - _Michael Somos_, Jun 04 2000
%F 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_, Aug 19 2003
%F 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_, Mar 03 2006
%F 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_, Apr 15 2006
%F G.f.: A(x) is x times the logarithmic derivative of A117209(x). - _Stuart Clary_, Apr 15 2006
%F Row sums of triangle A134842. - _Gary W. Adamson_, Nov 12 2007
%F G.f.: x/(1-x) = Sum_{n >= 1} a(n)*x^n/(1-x^n)^2. - _Paul D. Hanna_, Aug 16 2008
%F a(n) = phi(rad(n)) *(-1)^omega(n) = A000010(A007947(n)) *(-1)^A001221(n). - _Enrique Pérez Herrero_, Aug 24 2010
%F a(n) = Product_{i = 2..n} (1-i)^( (pi(i)-pi(i-1)) * floor( (cos(n*Pi/i))^2 ) ), where pi = A000720, Pi = A000796. - _Wesley Ivan Hurt_, May 24 2013
%F a(n) = -limit of zeta(s)*(Sum_{d divides n} moebius(d)/exp(d)^(s-1)) as s->1 for n>1. - _Mats Granvik_, Jul 31 2013
%F a(n) = Sum_{d divides n} mu(d)*rad(d), where rad is A007947. - _Enrique Pérez Herrero_, May 29 2014
%F Conjecture for n>1: Let n = 2^(A007814(n))*m = 2^(ruler(n))*odd_part(n), where m = A000265(n), then a(n) = (-1)^(m=n)*(0+Sum_{i=1..m and gcd(i,m)=1} (4*min(i,m-i)-m)) = (-1)^(m<n)*(1+Sum_{i=1..m and gcd(i,m)>1} (4*min(i,m-i)-m)). - _I. V. Serov_, May 02 2017
%F a(n) = (-1)^A001221(n) * A173557(n). - _R. J. Mathar_, Nov 02 2017
%F a(1) = 1; for n > 1, a(n) = (1-A020639(n)) * a(A028234(n)), because multiplicative with a(p^e) = (1-p). - _Antti Karttunen_, Nov 28 2017
%F a(n) = 1 - Sum_{d|n, d > 1} d*a(n/d). - _Ilya Gutkovskiy_, Apr 26 2019
%F From _Richard L. Ollerton_, May 07 2021: (Start)
%F For n>1, Sum_{k=1..n} a(gcd(n,k)) = 0.
%F For n>1, Sum_{k=1..n} a(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)) = 0. (End)
%F a(n) = rad(n)*(-1)^omega(n)*phi(n)/n = A062953(n)*A000010(n)/n. - _Amrit Awasthi_, Jan 30 2022
%F a(n) = mu(n)*phi(n) = A008683(n)*A000010(n) whenever n is squarefree. - _Amrit Awasthi_, Feb 03 2022
%F From _Peter Bala_, Jan 24 2024: (Start)
%F a(n) = Sum_{d divides n} core(d)*mu(d). Cf. Comment by _Benoit Cloitre_, dated Apr 06 2002.
%F a(n) = Sum_{d|n, e|n} n/gcd(d, e) * mu(n/d) * mu(n/e) (the sum is a multiplicative function of n by Tóth, and takes the value 1 - p for n = p^e, a prime power). (End)
%F From _Peter Bala_, Feb 01 2024: (Start)
%F G.f. Sum_{n >= 1} (2*n-1)*moebius(2*n-1)*x^(2*n-1)/(1 + x^(2n-1)).
%F a(n) = (-1)^(n+1) * Sum_{d divides n, d odd} d*moebius(d). (End)
%e x - x^2 - 2*x^3 - x^4 - 4*x^5 + 2*x^6 - 6*x^7 - x^8 - 2*x^9 + 4*x^10 - ...
%p A023900 := n -> mul(1-i,i=numtheory[factorset](n)); # _Peter Luschny_, Oct 26 2010
%t a[ n_] := If[ n < 1, 0, Sum[ d MoebiusMu @ d, { d, Divisors[n]}]] (* _Michael Somos_, Jul 18 2011 *)
%t Array[ Function[ n, 1/Plus @@ Map[ #*MoebiusMu[ # ]/EulerPhi[ # ]&, Divisors[ n ] ] ], 90 ]
%t nmax = 81; Drop[ CoefficientList[ Series[ Sum[ MoebiusMu[k] k x^k/(1 - x^k), {k, 1, nmax} ], {x, 0, nmax} ], x ], 1 ] (* _Stuart Clary_, Apr 15 2006 *)
%t t[n_, 1] = 1; t[1, k_] = 1; t[n_, k_] := t[n, k] = If[n < k, If[n > 1 && k > 1, Sum[-t[k - i, n], {i, 1, n - 1}], 0], If[n > 1 && k > 1, Sum[-t[n - i, k], {i, 1, k - 1}], 0]]; Table[t[n, n], {n, 36}] (* _Mats Granvik_, _Robert G. Wilson v_, Jun 25 2011 *)
%t Table[DivisorSum[m, # MoebiusMu[#] &], {m, 90}] (* _Jan Mangaldan_, Mar 15 2013 *)
%t f[p_, e_] := (1 - p); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Oct 14 2020 *)
%o (PARI) {a(n) = direuler( p=2, n, (1 - p*X) / (1 - X))[n]}
%o (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, d * moebius(d)))} /* _Michael Somos_, Jul 18 2011 */
%o (PARI) a(n)=sumdivmult(n,d, d*moebius(d)) \\ _Charles R Greathouse IV_, Sep 09 2014
%o (Haskell)
%o a023900 1 = 1
%o a023900 n = product $ map (1 -) $ a027748_row n
%o -- _Reinhard Zumkeller_, Jun 01 2015
%o (Python)
%o from sympy import divisors, mobius
%o def a(n): return sum([d*mobius(d) for d in divisors(n)]) # _Indranil Ghosh_, Apr 29 2017
%o (Python)
%o from math import prod
%o from sympy import primefactors
%o def A023900(n): return prod(1-p for p in primefactors(n)) # _Chai Wah Wu_, Sep 08 2023
%o (Scheme, with memoization-macro definec) (definec (A023900 n) (if (= 1 n) 1 (* (- 1 (A020639 n)) (A023900 (A028234 n))))) ;; _Antti Karttunen_, Nov 28 2017
%Y Cf. A000010, A007913, A023898, A117209, A134842.
%Y Moebius transform is A055615.
%Y Cf. A027748, A173557 (gives the absolute values), A295876.
%Y Cf. A253905 (Dgf at s=3).
%K sign,easy,nice,mult
%O 1,3
%A _Olivier Gérard_
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