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A339684 a(n) = Sum_{d|n} 4^(d-1).

%I A339684 #24 Jun 26 2024 02:05:43
%S A339684 1,5,17,69,257,1045,4097,16453,65553,262405,1048577,4195413,16777217,
%T A339684 67112965,268435729,1073758277,4294967297,17179935765,68719476737,
%U A339684 274878169413,1099511631889,4398047559685,17592186044417,70368748389461,281474976710913
%N A339684 a(n) = Sum_{d|n} 4^(d-1).
%H A339684 Seiichi Manyama, <a href="/A339684/b339684.txt">Table of n, a(n) for n = 1..1000</a>
%F A339684 G.f.: Sum_{k>=1} x^k / (1 - 4*x^k).
%F A339684 G.f.: Sum_{k>=1} 4^(k-1) * x^k / (1 - x^k).
%F A339684 a(n) ~ 4^(n-1). - _Vaclav Kotesovec_, Jun 05 2021
%t A339684 Table[Sum[4^(d - 1), {d, Divisors[n]}], {n, 1, 25}]
%t A339684 nmax = 25; CoefficientList[Series[Sum[x^k/(1 - 4 x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
%o A339684 (PARI) a(n) = sumdiv(n, d, 4^(d-1)); \\ _Michel Marcus_, Dec 13 2020
%o A339684 (Magma)
%o A339684 A339684:= func< n | (&+[4^(d-1): d in Divisors(n)]) >;
%o A339684 [A339684(n): n in [1..40]]; // _G. C. Greubel_, Jun 25 2024
%o A339684 (SageMath)
%o A339684 def A339684(n): return sum(4^(k-1) for k in (1..n) if (k).divides(n))
%o A339684 [A339684(n) for n in range(1,41)] # _G. C. Greubel_, Jun 25 2024
%Y A339684 Cf. A000302, A295505.
%Y A339684 Sums of the form Sum_{d|n} q^(d-1): A034729 (q=2), A034730 (q=3), A113999 (q=10), this sequence (q=4), A339685 (q=5), A339686 (q=6), A339687 (q=7), A339688 (q=8), A339689 (q=9).
%K A339684 nonn
%O A339684 1,2
%A A339684 _Ilya Gutkovskiy_, Dec 12 2020