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%I #33 Jan 21 2022 07:43:00
%S 1,2,4,5,8,10,16,20,32,40,64,80,128,160,256,320,512,640,1024,1280,
%T 2048,2560,4096,5120,8192,10240,16384,20480,32768,40960,65536,81920,
%U 131072,163840,262144,327680,524288,655360,1048576,1310720,2097152
%N Numbers of the form 2^k or 5*2^k.
%C The subset {a(1),...,a(2k)} together with a(2k+2) is the set of proper divisors of 5*2^k.
%C For a(n)>4: number of vertices of complete graphs that can be properly edge-colored in such a way that the edges can be partitioned into edge disjoint multicolored isomorphic spanning trees.
%C (Editor's note: The following 3 comments are equivalent.)
%C From _Wouter Meeussen_, Apr 10 2005: This appears to be the same sequence as "Numbers n such that n^2 is not the sum of three nonzero squares". _Don Reble_ and Paul Pollack respond: Yes, that is correct.
%C Also numbers k such that k^2=a^2+b^2+c^2 has no solutions in the positive integers a, b and c. - _Wouter Meeussen_, Apr 20 2005
%C The only natural numbers which cannot be the lengths of an interior diagonal of a cuboid with natural edges. - _Michael Somos_, Mar 02 2004
%D Wacław Sierpiński, Pythagorean triangles, Dover Publications, Inc., Mineola, NY, 2003, p. 101, MR2002669.
%H Gregory Constantine, <a href="https://doi.org/10.46298/dmtcs.306">Multicolored isomorphic spanning trees in complete graphs</a>, Discrete Mathematics and Theoretical Computer Science, Vol. 5 (2002), pp. 121-126.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,2).
%F a(1)=1, a(2)=2, a(3)=4, for n>=0, a(2n+3) = 4*2^n, a(2n+4) = 5*2^n.
%F Recurrence: for n>4, a(n) = 2a(n-2).
%F G.f.: x*(1+x)*(1+x+x^2)/(1-2x^2).
%F Sum_{n>=1} 1/a(n) = 12/5. - _Amiram Eldar_, Jan 21 2022
%t With[{c=2^Range[0,30]},Union[Join[c,5c]]] (* _Harvey P. Dale_, Jul 15 2012 *)
%Y Cf. A029744, A029745.
%Y Union of A000079 and A020714.
%Y Complement of A005767.
%K nonn,easy
%O 1,2
%A _Ralf Stephan_, Jun 01 2004
%E Edited by _T. D. Noe_ and _M. F. Hasler_, Nov 12 2010
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