A094958 Numbers of the form 2^k or 5*2^k.

1, 2, 4, 5, 8, 10, 16, 20, 32, 40, 64, 80, 128, 160, 256, 320, 512, 640, 1024, 1280, 2048, 2560, 4096, 5120, 8192, 10240, 16384, 20480, 32768, 40960, 65536, 81920, 131072, 163840, 262144, 327680, 524288, 655360, 1048576, 1310720, 2097152

Offset: 1

Author

Ralf Stephan, Jun 01 2004

Keywords

• nonn
• easy

Comments

The subset {a(1),...,a(2k)} together with a(2k+2) is the set of proper divisors of 5*2^k.

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.

(Editor's note: The following 3 comments are equivalent.)

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.

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

The only natural numbers which cannot be the lengths of an interior diagonal of a cuboid with natural edges. - Michael Somos, Mar 02 2004

References

• Wacław Sierpiński, Pythagorean triangles, Dover Publications, Inc., Mineola, NY, 2003, p. 101, MR2002669.

Links

• Gregory Constantine, Multicolored isomorphic spanning trees in complete graphs, Discrete Mathematics and Theoretical Computer Science, Vol. 5 (2002), pp. 121-126.
• Index entries for linear recurrences with constant coefficients, signature (0,2).

Crossrefs

Cf. A029744, A029745.

Union of A000079 and A020714.

Complement of A005767.

Programs

Mathematica

With[{c=2^Range[0,30]},Union[Join[c,5c]]] (* Harvey P. Dale, Jul 15 2012 *)

Python

def A094958(n): return 1<>1)+1 if n&1 else 5<<((n>>1)-2) # Chai Wah Wu, Feb 14 2025

Formula

a(1)=1, a(2)=2, a(3)=4, for n>=0, a(2n+3) = 4*2^n, a(2n+4) = 5*2^n.

Recurrence: for n>4, a(n) = 2a(n-2).

G.f.: x*(1+x)*(1+x+x^2)/(1-2x^2).

Sum_{n>=1} 1/a(n) = 12/5. - Amiram Eldar, Jan 21 2022

Extensions

Edited by T. D. Noe and M. F. Hasler, Nov 12 2010