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%I #30 Feb 04 2023 14:14:17
%S 1,4,9,20,37,76,141,280,541,1072,2097,4192,8289,16548,32953,65860,
%T 131397,262764,524909,1049736,2098381,4196560,8390865,16781696,
%U 33558929,67117460,134226585,268452580,536888037,1073775900,2147517725,4295034280,8590002605,17180002736,34359872001,68719743792
%N a(n) = Sum_{k=1..n} floor(n/k)*2^(k-1).
%C This is the "floor transform" of the powers of 2.
%H Matthew House, <a href="/A268235/b268235.txt">Table of n, a(n) for n = 1..3305</a>
%F a(n) ~ 2^n. - _Vaclav Kotesovec_, May 28 2021
%F From _Seiichi Manyama_, May 29 2021: (Start)
%F a(n) = Sum_{k=1..n} Sum_{d|k} 2^(d-1).
%F G.f.: (1/(1 - x)) * Sum_{k>=1} x^k/(1 - 2*x^k).
%F G.f.: (1/(1 - x)) * Sum_{k>=1} 2^(k-1) * x^k/(1 - x^k). (End)
%F a(n) = Sum_{k=1..n} (2^floor(n/k) - 1). - _Ridouane Oudra_, Feb 03 2023
%p # floor transform of a sequence
%p ft:=proc(a) local b,n,j,k; b:=[];
%p for n from 1 to nops(a) do j:=add(a[k]*floor(n/k),k=1..n); b:=[op(b),j]; od;
%p b; end:
%p ft([seq(2^i,i=0..50)]);
%t Table[Sum[Floor[n/k] 2^(k - 1), {k, n}], {n, 36}] (* _Michael De Vlieger_, Feb 12 2017 *)
%o (PARI) a(n) = sum(k=1, n, (n\k)*2^(k-1)); \\ _Michel Marcus_, Feb 11 2017
%o (PARI) a(n) = sum(k=1, n, sumdiv(k, d, 2^(d-1))); \\ _Seiichi Manyama_, May 29 2021
%o (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-2*x^k))/(1-x)) \\ _Seiichi Manyama_, May 29 2021
%o (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, 2^(k-1)*x^k/(1-x^k))/(1-x)) \\ _Seiichi Manyama_, May 29 2021
%Y First differences give A034729. Cf. A000079.
%K nonn
%O 1,2
%A _Benoit Cloitre_ and _N. J. A. Sloane_, Feb 05 2016
%E Definition corrected by _Matthew House_, Feb 11 2017
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