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A057660
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Sum_{k=1..n} n/GCD(n,k).
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29
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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
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OFFSET
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1,2
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COMMENTS
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Also sum of the orders of the elements in a cyclic group with n elements. - Avi Peretz (njk(AT)netvision.net.il), Mar 31 2001
Also inverse Moebius transform of EulerPhi[n^2].
Sequence is multiplicative: 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 E. (labos(AT)ana.sote.hu), Nov 21 2001
Multiplicative with a(p^e) = (p^(2*e+1)+1)/(p+1).
Equals A054522 * [1, 2, 3,...]. - Gary W. Adamson, May 21 2008
Row sums of triangle A054531. [Reinhard Zumkeller, Aug 12 2009]
a(n) <= (n-1)*n + 1, with equality if and only if n is noncomposite. - Daniel Forgues, April 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
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REFERENCES
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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)
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.
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.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..1000
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FORMULA
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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) = sum_{d|n} d*A000010(d). - Enrique Pérez Herrero, Jul 12 2010
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MATHEMATICA
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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 *)
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PROG
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(PARI) a(n)=if(n<1, 0, sumdiv(n, d, d*eulerphi(d)))
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CROSSREFS
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Cf. A018804, A051193, A057661, A001157, A000290, A000203, A065764, A054522, A226512.
Sequence in context: A178881 A095352 A061258 * A130972 A151923 A187264
Adjacent sequences: A057657 A057658 A057659 * A057661 A057662 A057663
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KEYWORD
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easy,nice,nonn,mult,changed
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AUTHOR
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Henry Gould, Oct 15 2000
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EXTENSIONS
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More terms from James A. Sellers, Oct 16 2000
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STATUS
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approved
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