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 A057660 a(n) = Sum_{k=1..n} n/gcd(n,k). 48
 1, 3, 7, 11, 21, 21, 43, 43, 61, 63, 111, 77, 157, 129, 147, 171, 273, 183, 343, 231, 301, 333, 507, 301, 521, 471, 547, 473, 813, 441, 931, 683, 777, 819, 903, 671, 1333, 1029, 1099, 903, 1641, 903, 1807, 1221, 1281, 1521, 2163, 1197, 2101, 1563, 1911, 1727 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Also sum of the orders of the elements in a cyclic group with n elements, i.e., row sums of A054531. - Avi Peretz (njk(AT)netvision.net.il), Mar 31 2001 Also inverse Moebius transform of EulerPhi(n^2), A002618. Sequence is multiplicative with a(p^e) = (p^(2*e+1)+1)/(p+1). Example: a(10) = a(2)*a(5) = 3*21 = 63. The lowest outliers, such that a(n)/((n-1)*n + 1) is a record low, seems to be given, except for 4, by A051451(n, n >= 3) = {6, 12, 60, 420, 840, 2520, 27720, ...}. If true, is there a proof? - Daniel Forgues, May 04 2013 a(n) is the number of pairs (a, b) such that the equation ax = b is solvable in the ring (Zn, +, x). See the Mathematical Reflections link. - Michel Marcus, Jan 07 2017 REFERENCES David M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston MA, 1976, p. 152. H. W. Gould and Temba Shonhiwa, Functions of GCD's and LCM's, Indian J. Math. (Allahabad), Vol. 39, No. 1 (1997), pp. 11-35. H. W. Gould and Temba Shonhiwa, A generalization of Cesaro's function and other results, Indian J. Math. (Allahabad), Vol. 39, No. 2 (1997), pp. 183-194. LINKS Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe) H. Amiri and S. M. Jafarian Amiri, Sum of element orders on finite groups of the same order, J. Algebra Appl. 10 (2011), no. 2, 187--190. MR2795731 (2012d:20050) Habib Amiri, S. M. Jafarian Amiri and I. M. Isaacs, Sums of element orders in finite groups Comm. Algebra 37 (2009), no. 9, 2978--2980. MR2554185 (2010i:20022) Miriam Mahannah El-Farrah, Expectation Numbers of Cyclic Groups, MS Thesis, Western Kentucky University, August 2015. Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 154-5. Walther Janous, Problem 10829, Amer. Math. Monthly, 107 (2000), p. 753. Yadollah Marefat, Ali Iranmanesh and Abolfazl Tehranian, On the sum of element orders of finite simple groups, J. Algebra Applications, 12 (2013), #1350026. Mathematical Reflections, Solution to Problem U338, Issue 4, 2015, p 17. Joachim von zur Gathen, Arnold Knopfmacher, Florian Luca, Lutz G. Lucht and Igor E. Shparlinski, Average order of cyclic groups, J. Théorie Nombres Bordeaux 16 (1) (2004) 107-123. FORMULA a(n) = Sum_{d|n} d*A000010(d) = Sum_{d|n} d*A054522(n,d), sum of d times phi(d) for all divisors d of n, where phi is Euler's phi function. a(n) = sigma_2(n^2)/sigma_1(n^2) = A001157(A000290(n))/A000203(A000290(n)) = A001157(A000290(n))/A065764(n). - Labos Elemer, Nov 21 2001 a(n) = Sum_{d|n} A000010(d^2). - Enrique Pérez Herrero, Jul 12 2010 a(n) <= (n-1)*n + 1, with equality if and only if n is noncomposite. - Daniel Forgues, Apr 30 2013 G.f.: Sum_{n >= 1} n*phi(n)*x^n/(1 - x^n) = x + 3*x^2 + 7*x^3 + 11*x^4 + .... Dirichlet g.f.: sum {n >= 1} a(n)/n^s = zeta(s)*zeta(s-2)/zeta(s-1) for Re s > 3.  Cf. A078747 and A176797. - Peter Bala, Dec 30 2013 a(n) = Sum_{i=1..n} numerator(n/i). - Wesley Ivan Hurt, Feb 26 2017 L.g.f.: -log(Product_{k>=1} (1 - x^k)^phi(k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 21 2018 From Richard L. Ollerton, May 10 2021: (Start) a(n) = Sum_{k=1..n} lcm(n,k)/k. a(n) = Sum_{k=1..n} gcd(n,k)*phi(gcd(n,k))/phi(n/gcd(n,k)). (End) From Vaclav Kotesovec, Jun 13 2021: (Start) Sum_{k=1..n} a(k)/k ~ 3*zeta(3)*n^2/Pi^2. Sum_{k=1..n} k^2/a(k) ~ A345294 * n. Sum_{k=1..n} k*A000010(k)/a(k) ~ A345295 * n. (End) MATHEMATICA Table[ DivisorSigma[ 2, n^2 ] / DivisorSigma[ 1, n^2 ], {n, 1, 128} ] Table[Total[Denominator[Range[n]/n]], {n, 55}] (* Alonso del Arte, Oct 07 2011 *) f[p_, e_] := (p^(2*e + 1) + 1)/(p + 1); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Nov 21 2020 *) PROG (PARI) a(n)=if(n<1, 0, sumdiv(n, d, d*eulerphi(d))) (PARI) a(n)=sumdivmult(n, d, eulerphi(d)*d) \\ Charles R Greathouse IV, Sep 09 2014 (Haskell) a057660 n = sum \$ map (div n) \$ a050873_row n -- Reinhard Zumkeller, Nov 25 2013 CROSSREFS Cf. A000010, A000203, A000290, A001157, A018804, A050873, A051193, A054522, A057661, A061255, A065764, A078747, A174405, A176797, A226512. Sequence in context: A244001 A333695 A061258 * A130972 A344483 A151923 Adjacent sequences:  A057657 A057658 A057659 * A057661 A057662 A057663 KEYWORD easy,nice,nonn,mult,changed AUTHOR Henry Gould, Oct 15 2000 EXTENSIONS More terms from James A. Sellers, Oct 16 2000 STATUS approved

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Last modified June 14 07:10 EDT 2021. Contains 345018 sequences. (Running on oeis4.)