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A053635 a(n) = Sum_{d|n} phi(d)*2^(n/d).

%I A053635 #44 Sep 08 2022 08:45:00
%S A053635 0,2,6,12,24,40,84,140,288,540,1080,2068,4224,8216,16548,32880,65856,
%T A053635 131104,262836,524324,1049760,2097480,4196412,8388652,16782048,
%U A053635 33554600,67117128,134218836,268452240,536870968,1073777040,2147483708,4295033472,8589938808
%N A053635 a(n) = Sum_{d|n} phi(d)*2^(n/d).
%C A053635 Dirichlet convolution of phi(n) and 2^n. - _Richard L. Ollerton_, May 06 2021
%H A053635 Seiichi Manyama, <a href="/A053635/b053635.txt">Table of n, a(n) for n = 0..3321</a>
%H A053635 James East and Ron Niles, <a href="https://arxiv.org/abs/1710.11245">Integer polygons of given perimeter</a>, arXiv:1710.11245 [math.CO], 2017.
%H A053635 James East and Ron Niles, <a href="https://doi.org/10.1017/S0004972718001612">Integer polygons of given perimeter</a>, Bull. Aust. Math. Soc. 100(1) (2019), 131-147.
%H A053635 T. Pisanski, D. Schattschneider, and B. Servatius, <a href="http://www.jstor.org/stable/27642932">Applying Burnside's lemma to a one-dimensional Escher problem</a>, Math. Mag., 79 (2006), 167-180. See v(n).
%F A053635 a(n) = n * A000031(n).
%F A053635 a(n) = Sum_{k=1..n} 2^gcd(n,k). - _Ilya Gutkovskiy_, Apr 16 2021
%F A053635 a(n) = Sum_{k=1..n} 2^(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - _Richard L. Ollerton_, May 06 2021
%p A053635 with(numtheory); A053685:=n->add( phi(n/d)*2^d, d in divisors(n)); # _N. J. A. Sloane_, Nov 21 2009
%t A053635 a[0] = 0; a[n_] := Sum[EulerPhi[d] 2^(n/d), {d, Divisors[n]}];
%t A053635 Table[a[n], {n, 0, 31}] (* _Jean-François Alcover_, Aug 30 2018 *)
%o A053635 (PARI) a(n) = if (n, sumdiv(n, d, eulerphi(d)*2^(n/d)), 0); \\ _Michel Marcus_, Sep 20 2017
%o A053635 (Magma) [0] cat  [&+[EulerPhi(d)*2^(n div d): d in Divisors(n)]: n in [1..40]]; // _Vincenzo Librandi_, Jul 20 2019
%Y A053635 Cf. A000031, A053634, A053636. Twice A034738.
%Y A053635 Column k=2 of A185651.
%K A053635 nonn
%O A053635 0,2
%A A053635 _N. J. A. Sloane_, Mar 23 2000