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A268235
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a(n) = Sum_{k=1..n} floor(n/k)*2^(k-1).
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10
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1, 4, 9, 20, 37, 76, 141, 280, 541, 1072, 2097, 4192, 8289, 16548, 32953, 65860, 131397, 262764, 524909, 1049736, 2098381, 4196560, 8390865, 16781696, 33558929, 67117460, 134226585, 268452580, 536888037, 1073775900, 2147517725, 4295034280, 8590002605, 17180002736, 34359872001, 68719743792
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OFFSET
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1,2
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COMMENTS
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This is the "floor transform" of the powers of 2.
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LINKS
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FORMULA
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a(n) = Sum_{k=1..n} Sum_{d|k} 2^(d-1).
G.f.: (1/(1 - x)) * Sum_{k>=1} x^k/(1 - 2*x^k).
G.f.: (1/(1 - x)) * Sum_{k>=1} 2^(k-1) * x^k/(1 - x^k). (End)
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MAPLE
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# floor transform of a sequence
ft:=proc(a) local b, n, j, k; b:=[];
for n from 1 to nops(a) do j:=add(a[k]*floor(n/k), k=1..n); b:=[op(b), j]; od;
b; end:
ft([seq(2^i, i=0..50)]);
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MATHEMATICA
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PROG
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(PARI) a(n) = sum(k=1, n, (n\k)*2^(k-1)); \\ Michel Marcus, Feb 11 2017
(PARI) a(n) = sum(k=1, n, sumdiv(k, d, 2^(d-1))); \\ Seiichi Manyama, May 29 2021
(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
(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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