A130887 Inverse Moebius transform of the Mersenne numbers: a(n) = Sum_{d|n} (2^d - 1).

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

Cf. A001047.

Cf. A027750, A000225, A038199.

Sequence in context: A280114 A162362 A274817 * A049933 A301746 A163318

Adjacent sequences: A130884 A130885 A130886 * A130888 A130889 A130890

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