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A328408 G.f. A(x) satisfies: A(x) = A(x^2) + x * (1 + 4*x + x^2) / (1 - x)^4.

%I A328408 #20 Oct 23 2023 02:01:13
%S A328408 1,9,27,73,125,243,343,585,729,1125,1331,1971,2197,3087,3375,4681,
%T A328408 4913,6561,6859,9125,9261,11979,12167,15795,15625,19773,19683,25039,
%U A328408 24389,30375,29791,37449,35937,44217,42875,53217,50653,61731,59319,73125,68921,83349,79507,97163,91125
%N A328408 G.f. A(x) satisfies: A(x) = A(x^2) + x * (1 + 4*x + x^2) / (1 - x)^4.
%H A328408 Amiram Eldar, <a href="/A328408/b328408.txt">Table of n, a(n) for n = 1..10000</a>
%F A328408 G.f.: Sum_{k>=0} x^(2^k) * (1 + 4*x^(2^k) + x^(2^(k+1))) / (1 - x^(2^k))^4.
%F A328408 G.f.: (1/7) * Sum_{k>=1} J_3(2*k) * x^k / (1 - x^k), where J_3() is the Jordan function (A059376).
%F A328408 Dirichlet g.f.: zeta(s-3) / (1 - 2^(-s)).
%F A328408 a(2*n) = a(n) + 8*n^3, a(2*n+1) = (2*n + 1)^3.
%F A328408 a(n) = Sum_{d|n} A209229(n/d) * d^3.
%F A328408 Product_{n>=1} (1 + x^n)^a(n) = g.f. for A023872.
%F A328408 Sum_{k=1..n} a(k) ~ 4*n^4/15. - _Vaclav Kotesovec_, Oct 15 2019
%F A328408 Multiplicative with a(2^e) = (8^(e+1)-1)/7, and a(p^e) = p^(3*e) for an odd prime p. - _Amiram Eldar_, Oct 23 2023
%t A328408 nmax = 45; CoefficientList[Series[Sum[x^(2^k) (1 + 4 x^(2^k) + x^(2^(k + 1)))/(1 - x^(2^k))^4, {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x] // Rest
%t A328408 a[n_] := If[EvenQ[n], a[n/2] + n^3, n^3]; Table[a[n], {n, 1, 45}]
%t A328408 Table[DivisorSum[n, Boole[IntegerQ[Log[2, n/#]]] #^3 &], {n, 1, 45}]
%t A328408 f[p_, e_] :=p^(3*e); f[2, e_] := (8^(e+1)-1)/7; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Oct 23 2023 *)
%o A328408 (Magma) [n eq 1 select 1 else IsOdd(n) select n^3 else Self(n div 2)+n^3 :n in [1..45]]; // _Marius A. Burtea_, Oct 15 2019
%Y A328408 Cf. A000578, A001511, A016755, A023872, A059376, A129527, A209229, A328407.
%K A328408 nonn,easy,mult
%O A328408 1,2
%A A328408 _Ilya Gutkovskiy_, Oct 14 2019