A069097 Moebius transform of A064987, n*sigma(n).

1, 5, 11, 22, 29, 55, 55, 92, 105, 145, 131, 242, 181, 275, 319, 376, 305, 525, 379, 638, 605, 655, 551, 1012, 745, 905, 963, 1210, 869, 1595, 991, 1520, 1441, 1525, 1595, 2310, 1405, 1895, 1991, 2668, 1721, 3025, 1891, 2882, 3045, 2755, 2255, 4136, 2737

Offset

1,2

Comments

Equals A127569 * [1, 2, 3, ...]. - Gary W. Adamson, Jan 19 2007

Equals row sums of triangle A143309 and of triangle A143312. - Gary W. Adamson, Aug 06 2008

Dirichlet convolution of A000290 and A000010 (see Jovovic formula). - R. J. Mathar, Feb 03 2011

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Wenchang Chu, Jordan's totient function and trigonometric sums, Studia Scientiarum Mathematicarum Hungarica 57 (1), 40-53 (2020).

Nikolai Osipov, Problem 12003, Amer. Math. Monthly, 124(8) (2017), p. 754.

Formula

a(n) = Sum_{d|n} d^2*phi(n/d). - Vladeta Jovovic, Jul 31 2002

a(n) = Sum_{k=1..n} gcd(n, k)^2. - Vladeta Jovovic, Aug 27 2003

Dirichlet g.f.: zeta(s-2)*zeta(s-1)/zeta(s). - R. J. Mathar, Feb 03 2011

a(n) = n*Sum_{d|n} J_2(d)/d, where J_2 is A007434. - Enrique Pérez Herrero, Feb 25 2012.

G.f.: Sum_{n >= 1} phi(n)*(x^n + x^(2*n))/(1 - x^n)^3 = x + 5*x^2 + 11*x^3 + 22*x^4 + .... - Peter Bala, Dec 30 2013

Multiplicative with a(p^e) = p^(e-1)*(p^e*(p+1)-1). - R. J. Mathar, Jun 23 2018

Sum_{k=1..n} a(k) ~ Pi^2 * n^3 / (18*zeta(3)). - Vaclav Kotesovec, Sep 18 2020

a(n) = Sum_{k=1..n} (n/gcd(n,k))^2*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021

From Peter Bala, Dec 26 2023: (Start)

For n odd, a(n) = Sum_{k = 1..n} gcd(k,n)/cos(k*Pi/n)^2 (see Osipov and also Chu, p. 51).

It appears that for n odd, Sum_{k = 1..n} (-1)^(k+1)*gcd(k,n)/cos(k*Pi/n)^2 = n. (End)

a(n) = Sum_{1 <= i, j <= n} gcd(i, j, n). Cf. A360428. - Peter Bala, Jan 16 2024

Mathematica

A069097[n_]:=n^2*Plus @@((EulerPhi[#]/#^2)&/@ Divisors[n]); Array[A069097, 100] (* Enrique Pérez Herrero, Feb 25 2012 *)
f[p_, e_] := p^(e-1)*(p^e*(p+1)-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2020 *)

Prog

(PARI) for(n=1, 100, print1((sumdiv(n, k, k*sigma(k)*moebius(n/k))), ", "))

Crossrefs

Column 2 of A343510.

Cf. A018804, A033457, A068963, A064987, A127569, A143309, A143312, A360428.

For Sum_{k = 1..n} gcd(k,n)^m see A018804 (m = 1), A343497 (m = 3), A343498 (m = 4) and A343499 (m = 5).

Sequence in context: A296033 A296968 A184552 * A222548 A024921 A189978

Adjacent sequences: A069094 A069095 A069096 * A069098 A069099 A069100

Keyword

easy,nonn,mult

Author

Benoit Cloitre, Apr 05 2002

Status

approved