VOOZH about

URL: https://oeis.org/A094958

⇱ A094958 - OEIS

login
A094958
Numbers of the form 2^k or 5*2^k.
17
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
1,2
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.
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
MATHEMATICA
With[{c=2^Range[0, 30]}, Union[Join[c, 5c]]] (* Harvey P. Dale, Jul 15 2012 *)
PROG
(Python)
def A094958(n): return 1<<n-1 if n<4 else 1<<(n>>1)+1 if n&1 else 5<<((n>>1)-2) # Chai Wah Wu, Feb 14 2025
CROSSREFS
Union of A000079 and A020714.
Complement of A005767.
Sequence in context: A228939 A115831 A391737 * A018565 A018391 A018310
KEYWORD
nonn,easy
AUTHOR
Ralf Stephan, Jun 01 2004
EXTENSIONS
Edited by T. D. Noe and M. F. Hasler, Nov 12 2010
STATUS
approved