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A130887
Inverse Moebius transform of the Mersenne numbers: a(n) = Sum_{d|n} (2^d - 1).
3
1, 4, 8, 19, 32, 74, 128, 274, 519, 1058, 2048, 4184, 8192, 16514, 32806, 65809, 131072, 262728, 524288, 1049648, 2097286, 4196354, 8388608, 16781654, 33554463, 67117058, 134218246, 268451984, 536870912, 1073775718, 2147483648, 4295033104, 8589936646
OFFSET
1,2
LINKS
Enrique Pérez Herrero, Table of n, a(n) for n = 1..500
FORMULA
a(n) = Sum_{d|n} Sum_{k=1..d} C(d,k) = Sum_{d|n} (-1 + 2^d) = Sum_{d|n} 2^d - tau(n) = A055895(n) - A000010(n). - Enrique Pérez Herrero, Apr 14 2012
G.f.: Sum_{k>=1} (2^k - 1)*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 28 2017
a(n) = Sum_{i=1..n} 2^(i-1)*A135539(n,i). - Ridouane Oudra, Sep 19 2022
EXAMPLE
G.f. = x + 4*x^2 + 8*x^3 + 19*x^4 + 32*x^5 + 74*x^6 + 128*x^7 + 274*x^8 + ...
MATHEMATICA
A130887[n_]:=DivisorSum[n, Plus@@Table[Binomial[#, i], {i, 1, #}]&]; Array[A130887, 20] (* Enrique Pérez Herrero, Apr 14 2012 *)
a[ n_] := If[ n < 1, 0, DivisorSum[ n, 2^# - 1 &]]; (* Michael Somos, Jan 28 2017 *)
PROG
(Haskell)
a130887 = sum . map a000225 . a027750_row
-- Reinhard Zumkeller, Feb 17 2013
(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, 2^d-1))}; /* Michael Somos, Jan 28 2017 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Gary W. Adamson, Jun 07 2007
EXTENSIONS
New name from Enrique Pérez Herrero, Apr 14 2012
Name corrected by Michel Marcus, Sep 19 2022
STATUS
approved