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%I A343497 #52 Jun 24 2024 18:37:18
%S A343497 1,9,29,74,129,261,349,596,789,1161,1341,2146,2209,3141,3741,4776,
%T A343497 4929,7101,6877,9546,10121,12069,12189,17284,16145,19881,21321,25826,
%U A343497 24417,33669,29821,38224,38889,44361,45021,58386,50689,61893,64061,76884,68961,91089,79549,99234,101781
%N A343497 a(n) = Sum_{k=1..n} gcd(k, n)^3.
%H A343497 Seiichi Manyama, <a href="/A343497/b343497.txt">Table of n, a(n) for n = 1..10000</a>
%F A343497 a(n) = Sum_{d|n} phi(n/d) * d^3.
%F A343497 a(n) = Sum_{d|n} mu(n/d) * d * sigma_2(d).
%F A343497 G.f.: Sum_{k >= 1} phi(k) * x^k * (1 + 4*x^k + x^(2*k))/(1 - x^k)^4.
%F A343497 Dirichlet g.f.: zeta(s-1) * zeta(s-3) / zeta(s). - _Ilya Gutkovskiy_, Apr 18 2021
%F A343497 Sum_{k=1..n} a(k) ~ 45*zeta(3)*n^4 / (2*Pi^4). - _Vaclav Kotesovec_, May 20 2021
%F A343497 Multiplicative with a(p^e) = p^(e-1)*((p^2+p+1)*p^(2*e) - 1)/(p+1). - _Amiram Eldar_, Nov 22 2022
%F A343497 a(n) = Sum_{1 <= i, j, k <= n} gcd(i, j, k, n) = Sum_{d divides n} d * J_3(n/d), where the Jordan totient function J_3(n) = A059376(n). - _Peter Bala_, Jan 20 2024
%p A343497 with(numtheory):
%p A343497 seq(add(phi(n/d) * d^3, d in divisors(n)), n = 1..50); # _Peter Bala_, Jan 20 2024
%t A343497 a[n_] := Sum[GCD[k, n]^3, {k, 1, n}]; Array[a, 50] (* _Amiram Eldar_, Apr 18 2021 *)
%t A343497 f[p_, e_] := p^(e - 1)*((p^2 + p + 1)*p^(2*e) - 1)/(p + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* _Amiram Eldar_, Nov 22 2022 *)
%t A343497 A343497[n_]:= DivisorSum[n, #^3*EulerPhi[n/#] &]; Table[A343497[n], {n, 50}] (* _G. C. Greubel_, Jun 24 2024 *)
%o A343497 (PARI) a(n) = sum(k=1, n, gcd(k, n)^3);
%o A343497 (PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*d^3);
%o A343497 (PARI) a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, 2));
%o A343497 (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+4*x^k+x^(2*k))/(1-x^k)^4))
%o A343497 (Magma)
%o A343497 A343497:= func< n | (&+[d^3*EulerPhi(Floor(n/d)): d in Divisors(n)]) >;
%o A343497 [A343497(n): n in [1..50]]; // _G. C. Greubel_, Jun 24 2024
%o A343497 (SageMath)
%o A343497 def A343497(n): return sum(k^3*euler_phi(n/k) for k in (1..n) if (k).divides(n))
%o A343497 [A343497(n) for n in range(1,51)] # _G. C. Greubel_, Jun 24 2024
%Y A343497 Column 3 of A343510.
%Y A343497 Cf. A000010, A001157 (sigma_2(n)), A018804, A054610, A069097, A309323, A332517, A342423, A342433, A343498, A343499, A343513.
%K A343497 nonn,mult,easy
%O A343497 1,2
%A A343497 _Seiichi Manyama_, Apr 17 2021

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