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%I #19 Dec 17 2022 03:15:42
%S 1,2,4,4,5,8,7,8,13,10,11,16,13,14,20,16,17,26,19,20,28,22,23,32,25,
%T 26,40,28,29,40,31,32,44,34,35,52,37,38,52,40,41,56,43,44,65,46,47,64,
%U 49,50,68,52,53,80,55,56,76,58,59,80,61,62,91,64,65,88,67,68,92,70
%N Expansion of Sum_{k>=0} x^(3^k) / (1 - x^(3^k))^2.
%C Sum of divisors d of n such that n/d is power of 3.
%C Inverse Moebius transform of A195459.
%H Antti Karttunen, <a href="/A327625/b327625.txt">Table of n, a(n) for n = 1..20000</a>
%F G.f. A(x) satisfies: A(x) = A(x^3) + x/(1 - x)^2.
%F G.f.: Sum_{k>=1} phi(3*k) * x^k / (2 * (1 - x^k)), where phi = A000010.
%F a(n) = (1/2) * Sum_{d|n} phi(3*d).
%F From _Amiram Eldar_, Nov 17 2022: (Start)
%F Multiplicative with a(3^e) = (3^(e+1)-1)/2, and a(p^e) = p^e for p != 3.
%F Sum_{k=1..n} a(k) ~ (9/16) * n^2. (End)
%F Dirichlet g.f.: zeta(s-1)*(1+1/(3^s-1)). - _Amiram Eldar_, Dec 17 2022
%t nmax = 70; CoefficientList[Series[Sum[x^(3^k)/(1 - x^(3^k))^2, {k, 0, Floor[Log[3, nmax]] + 1}], {x, 0, nmax}], x] // Rest
%t a[n_] := DivisorSum[n, # &, IntegerQ[Log[3, n/#]] &]; Table[a[n], {n, 1, 70}]
%t a[n_] := 1/2 Sum[EulerPhi[3 d], {d, Divisors[n]}]; Table[a[n], {n, 1, 70}]
%o (PARI) A327625(n) = (n+sumdiv(n,d,my(b=0); if(isprimepower(n/d,&b)&&(3==b),d,0))); \\ _Antti Karttunen_, Sep 19 2019
%o (Magma) [(1/2)*&+[EulerPhi(3*d) :d in Divisors(n)]:n in [1..70]]; // _Marius A. Burtea_, Sep 19 2019
%Y Cf. A000010, A000244, A001651 (fixed points), A051064, A129527, A195459.
%K nonn,mult
%O 1,2
%A _Ilya Gutkovskiy_, Sep 19 2019
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