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%I #74 May 11 2024 05:02:31
%S 1,5,11,22,29,55,55,92,105,145,131,242,181,275,319,376,305,525,379,
%T 638,605,655,551,1012,745,905,963,1210,869,1595,991,1520,1441,1525,
%U 1595,2310,1405,1895,1991,2668,1721,3025,1891,2882,3045,2755,2255,4136,2737
%N Moebius transform of A064987, n*sigma(n).
%C Equals A127569 * [1, 2, 3, ...]. - _Gary W. Adamson_, Jan 19 2007
%C Equals row sums of triangle A143309 and of triangle A143312. - _Gary W. Adamson_, Aug 06 2008
%C Dirichlet convolution of A000290 and A000010 (see Jovovic formula). - _R. J. Mathar_, Feb 03 2011
%H Amiram Eldar, <a href="/A069097/b069097.txt">Table of n, a(n) for n = 1..10000</a>
%H Wenchang Chu, <a href="https://doi.org/10.1556/012.2020.57.1.1452">Jordan's totient function and trigonometric sums</a>, Studia Scientiarum Mathematicarum Hungarica 57 (1), 40-53 (2020).
%H Nikolai Osipov, <a href="https://doi.org/10.4169/amer.math.monthly.124.8.754">Problem 12003</a>, Amer. Math. Monthly, 124(8) (2017), p. 754.
%F a(n) = Sum_{d|n} d^2*phi(n/d). - _Vladeta Jovovic_, Jul 31 2002
%F a(n) = Sum_{k=1..n} gcd(n, k)^2. - _Vladeta Jovovic_, Aug 27 2003
%F Dirichlet g.f.: zeta(s-2)*zeta(s-1)/zeta(s). - _R. J. Mathar_, Feb 03 2011
%F a(n) = n*Sum_{d|n} J_2(d)/d, where J_2 is A007434. - _Enrique Pérez Herrero_, Feb 25 2012.
%F 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
%F Multiplicative with a(p^e) = p^(e-1)*(p^e*(p+1)-1). - _R. J. Mathar_, Jun 23 2018
%F Sum_{k=1..n} a(k) ~ Pi^2 * n^3 / (18*zeta(3)). - _Vaclav Kotesovec_, Sep 18 2020
%F 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
%F From _Peter Bala_, Dec 26 2023: (Start)
%F For n odd, a(n) = Sum_{k = 1..n} gcd(k,n)/cos(k*Pi/n)^2 (see Osipov and also Chu, p. 51).
%F It appears that for n odd, Sum_{k = 1..n} (-1)^(k+1)*gcd(k,n)/cos(k*Pi/n)^2 = n. (End)
%F a(n) = Sum_{1 <= i, j <= n} gcd(i, j, n). Cf. A360428. - _Peter Bala_, Jan 16 2024
%F Sum_{k=1..n} a(k)/k ~ Pi^2 * n^2 / (12*zeta(3)). - _Vaclav Kotesovec_, May 11 2024
%t A069097[n_]:=n^2*Plus @@((EulerPhi[#]/#^2)&/@ Divisors[n]); Array[A069097, 100] (* _Enrique Pérez Herrero_, Feb 25 2012 *)
%t 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 *)
%o (PARI) for(n=1,100,print1((sumdiv(n,k,k*sigma(k)*moebius(n/k))),","))
%Y Column 2 of A343510.
%Y Cf. A018804, A033457, A068963, A064987, A127569, A143309, A143312, A360428.
%Y For Sum_{k = 1..n} gcd(k,n)^m see A018804 (m = 1), A343497 (m = 3), A343498 (m = 4) and A343499 (m = 5).
%K easy,nonn,mult,changed
%O 1,2
%A _Benoit Cloitre_, Apr 05 2002
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