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A344815 a(n) = Sum_{k=1..n} floor(n/k) * 4^(k-1).

%I A344815 #27 Jun 28 2024 05:05:23
%S A344815 1,6,23,92,349,1394,5491,21944,87497,349902,1398479,5593892,22371109,
%T A344815 89484074,357919803,1431678080,5726645377,22906581142,91626057879,
%U A344815 366504227292,1466015859181,5864063418866,23456249463283,93824997852744,375299974563657,1501199898183502
%N A344815 a(n) = Sum_{k=1..n} floor(n/k) * 4^(k-1).
%C A344815 Partial sums of A339684.
%H A344815 Seiichi Manyama, <a href="/A344815/b344815.txt">Table of n, a(n) for n = 1..1000</a>
%F A344815 a(n) = Sum_{k=1..n} Sum_{d|k} 4^(d-1).
%F A344815 G.f.: (1/(1 - x)) * Sum_{k>=1} x^k/(1 - 4*x^k).
%F A344815 G.f.: (1/(1 - x)) * Sum_{k>=1} 4^(k-1) * x^k/(1 - x^k).
%F A344815 a(n) ~ 4^n / 3. - _Vaclav Kotesovec_, Jun 05 2021
%F A344815 a(n) = (1/3) * Sum_{k=1..n} (4^floor(n/k) - 1). - _Ridouane Oudra_, Feb 16 2023
%p A344815 seq(add(4^(k-1)*floor(n/k), k=1..n), n=1..60); # _Ridouane Oudra_, Feb 16 2023
%t A344815 a[n_] := Sum[4^(k - 1) * Quotient[n, k], {k, 1, n}]; Array[a, 30] (* _Amiram Eldar_, May 29 2021 *)
%o A344815 (PARI) a(n) = sum(k=1, n, n\k*4^(k-1));
%o A344815 (PARI) a(n) = sum(k=1, n, sumdiv(k, d, 4^(d-1)));
%o A344815 (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-4*x^k))/(1-x))
%o A344815 (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, 4^(k-1)*x^k/(1-x^k))/(1-x))
%o A344815 (Magma)
%o A344815 A344815:= func< n | (&+[Floor(n/j)*4^(j-1): j in [1..n]]) >;
%o A344815 [A344815(n): n in [1..40]]; // _G. C. Greubel_, Jun 27 2024
%o A344815 (SageMath)
%o A344815 def A344815(n): return sum((n//j)*4^(j-1) for j in range(1,n+1))
%o A344815 [A344815(n) for n in range(1,41)] # _G. C. Greubel_, Jun 27 2024
%Y A344815 Column k=4 of A344821.
%Y A344815 Sums of the form Sum_{k=1..n} q^(k-1)*floor(n/k): A344820 (q=-n), A344819 (q=-4), A344818 (q=-3), A344817 (q=-2), A059851 (q=-1), A006218 (q=1), A268235 (q=2), A344814 (q=3), this sequence (q=4), A344816 (q=5), A332533 (q=n).
%K A344815 nonn
%O A344815 1,2
%A A344815 _Seiichi Manyama_, May 29 2021