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

%I A130887 #45 Sep 19 2022 11:17:01
%S A130887 1,4,8,19,32,74,128,274,519,1058,2048,4184,8192,16514,32806,65809,
%T A130887 131072,262728,524288,1049648,2097286,4196354,8388608,16781654,
%U A130887 33554463,67117058,134218246,268451984,536870912,1073775718,2147483648,4295033104,8589936646
%N A130887 Inverse Moebius transform of the Mersenne numbers: a(n) = Sum_{d|n} (2^d - 1).
%H A130887 Enrique Pérez Herrero, <a href="/A130887/b130887.txt">Table of n, a(n) for n = 1..500</a>
%F A130887 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
%F A130887 G.f.: Sum_{k>=1} (2^k - 1)*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Jan 28 2017
%F A130887 a(n) = Sum_{i=1..n} 2^(i-1)*A135539(n,i). - _Ridouane Oudra_, Sep 19 2022
%e A130887 G.f. = x + 4*x^2 + 8*x^3 + 19*x^4 + 32*x^5 + 74*x^6 + 128*x^7 + 274*x^8 + ...
%t A130887 A130887[n_]:=DivisorSum[n,Plus@@Table[Binomial[#,i],{i,1,#}]&]; Array[A130887,20] (* _Enrique Pérez Herrero_, Apr 14 2012 *)
%t A130887 a[ n_] := If[ n < 1, 0, DivisorSum[ n, 2^# - 1 &]]; (* _Michael Somos_, Jan 28 2017 *)
%o A130887 (Haskell)
%o A130887 a130887 = sum . map a000225 . a027750_row
%o A130887 -- _Reinhard Zumkeller_, Feb 17 2013
%o A130887 (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, 2^d-1))}; /* _Michael Somos_, Jan 28 2017 */
%Y A130887 Cf. A001047.
%Y A130887 Cf. A027750, A000225, A038199.
%K A130887 nonn
%O A130887 1,2
%A A130887 _Gary W. Adamson_, Jun 07 2007
%E A130887 New name from _Enrique Pérez Herrero_, Apr 14 2012
%E A130887 Name corrected by _Michel Marcus_, Sep 19 2022