login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A057660 Sum_{k=1..n} n/GCD(n,k). 30
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.

Miriam Mahannah El-Farrah, Expectation Numbers of Cyclic Groups, MS Thesis, Western Kentucky University, August 2015; http://digitalcommons.wku.edu/cgi/viewcontent.cgi?article=2520&context=theses

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

AUTHOR

Henry Gould, Oct 15 2000

EXTENSIONS

More terms from James A. Sellers, Oct 16 2000

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified May 24 21:22 EDT 2016. Contains 273253 sequences.