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A080267 a(n) = Sum_{d divides n} d*2^(n-n/d).

%I A080267 #27 Dec 20 2022 10:38:13
%S A080267 1,5,13,41,81,257,449,1313,2497,6465,11265,33665,53249,143617,269313,
%T A080267 672257,1114113,3159041,4980737,13568001,23904257,57675777,96468993,
%U A080267 275980289,424673281,1090535425,1963720705,4823482369,7784628225
%N A080267 a(n) = Sum_{d divides n} d*2^(n-n/d).
%H A080267 Robert Israel, <a href="/A080267/b080267.txt">Table of n, a(n) for n = 1..3307</a>
%F A080267 G.f.: Sum_{k>=1} k*2^(k-1)*x^k/(1 - 2^(k-1)*x^k). - _N. J. A. Sloane_, Jun 04 2003
%F A080267 G.f.: Sum_{k>=1} x^k/(1 - (2 * x)^k)^2. - _Seiichi Manyama_, Dec 20 2022
%p A080267 oo := 40; s1 := add( k*2^(k-1)*x^k/(1-2^(k-1)*x^k),k=1..oo): s2 := series(s1,x,oo-1): s3 := seriestolist(%): A080267 := n->s3[n+1];
%t A080267 a[n_] := Sum[d*2^(n-n/d), {d, Divisors[n]}]; Array[a, 29] (* _Jean-François Alcover_, Mar 20 2014 *)
%o A080267 (PARI) a(n) = sumdiv(n, d, d*2^(n-n/d)); \\ _Michel Marcus_, Mar 20 2014
%o A080267 (PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-(2*x)^k)^2)) \\ _Seiichi Manyama_, Dec 20 2022
%Y A080267 Cf. A075900, A359112.
%K A080267 nonn
%O A080267 1,2
%A A080267 _Vladeta Jovovic_, Feb 11 2003