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A339688 a(n) = Sum_{d|n} 8^(d-1).

%I A339688 #19 Jun 26 2024 02:06:03
%S A339688 1,9,65,521,4097,32841,262145,2097673,16777281,134221833,1073741825,
%T A339688 8589967945,68719476737,549756076041,4398046515265,35184374186505,
%U A339688 281474976710657,2251799830495305,18014398509481985,144115188210078217,1152921504607109185
%N A339688 a(n) = Sum_{d|n} 8^(d-1).
%H A339688 Seiichi Manyama, <a href="/A339688/b339688.txt">Table of n, a(n) for n = 1..1000</a>
%F A339688 G.f.: Sum_{k>=1} x^k / (1 - 8*x^k).
%F A339688 G.f.: Sum_{k>=1} 8^(k-1) * x^k / (1 - x^k).
%F A339688 a(n) ~ 8^(n-1). - _Vaclav Kotesovec_, Jun 05 2021
%t A339688 Table[Sum[8^(d - 1), {d, Divisors[n]}], {n, 1, 21}]
%t A339688 nmax = 21; CoefficientList[Series[Sum[x^k/(1 - 8 x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
%o A339688 (PARI) a(n) = sumdiv(n, d, 8^(d-1)); \\ _Michel Marcus_, Dec 13 2020
%o A339688 (Magma)
%o A339688 A339688:= func< n | (&+[8^(d-1): d in Divisors(n)]) >;
%o A339688 [A339688(n): n in [1..40]]; // _G. C. Greubel_, Jun 25 2024
%o A339688 (SageMath)
%o A339688 def A339688(n): return sum(8^(k-1) for k in (1..n) if (k).divides(n))
%o A339688 [A339688(n) for n in range(1,41)] # _G. C. Greubel_, Jun 25 2024
%Y A339688 Column 8 of A308813.
%Y A339688 Cf. A001018, A320073.
%Y A339688 Sums of the form Sum_{d|n} q^(d-1): A034729 (q=2), A034730 (q=3), A113999 (q=10), A339684 (q=4), A339685 (q=5), A339686 (q=6), A339687 (q=7), this sequence (q=8), A339689 (q=9).
%K A339688 nonn
%O A339688 1,2
%A A339688 _Ilya Gutkovskiy_, Dec 12 2020