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A057660 Sum_{k=1..n} n/GCD(n,k). 29
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.

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) <= (n-1)*n + 1, with equality if and only if n is noncomposite. - Daniel Forgues, Apr 30 2013

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

REFERENCES

D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston MA, 1976, p. 152.

H. W. Gould and Temba Shonhiwa, Indian J. Math. (Allahabad), 39 (1997), 11-35.

H. W. Gould and Temba Shonhiwa, Indian J. Math. (Allahabad), 39 (1997), 183-194.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

Amiri, H.; Amiri, S. M. Jafarian, Sum of element orders on finite groups of the same order, J. Algebra Appl. 10 (2011), no. 2, 187--190. MR2795731 (2012d:20050)

Amiri, Habib; Jafarian Amiri, S. M.; Isaacs, I. M. Sums of element orders in finite groups Comm. Algebra 37 (2009), no. 9, 2978--2980. MR2554185 (2010i:20022)

Walther Janous, Problem 10829, Amer. Math. Monthly, 107 (2000), p. 753.

Y. Marefat et al., On the sum of element orders of finite simple groups, J. Algebra Applications, 12 (2013), #1350026.

J. von zur Gathen, A. Knopfmacher, F. Luca, L. G. Lucht, I. E. Shparlinski, Average order of cyclic groups, J. Theorie 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) = sum_{d|n} A000010(d^2). - Enrique Pérez Herrero, Jul 12 2010

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

MATHEMATICA

Table[ DivisorSigma[ 2, n^2 ]/[ DivisorSigma[ 1, n^2 ], {n, 1, 128} ] (* Gould *)

Table[Total[Denominator[Range[n]/n]], {n, 55}] (* Alonso del Arte, Oct 07 2011 *)

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. A018804, A051193, A057661, A001157, A000290, A000203, A065764, A054522, A226512, A050873.  A078747, A176797.

Sequence in context: A095352 A244001 A061258 * A130972 A151923 A187264

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 September 16 07:24 EDT 2014. Contains 246799 sequences.