login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A108345 The set N such that each positive integer can be written in the form s^2 + n, s>=0, n in N, in an even number of ways. 4
0, 1, 2, 3, 5, 7, 8, 9, 13, 17, 18, 23, 27, 29, 31, 32, 35, 37, 39, 41, 45, 47, 49, 50, 53, 55, 59, 61, 63, 71, 72, 73, 79, 81, 83, 87, 89, 91, 97, 98, 101, 103, 107, 109, 113, 115, 117, 121, 127, 128, 137, 139, 149, 151, 153, 157, 159, 162, 167, 171, 173, 181, 183, 191 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
This set is conjectured to have zero density. The only even values are the numbers of the form 2n^2. See the paper by Cooper, Eichhorn and O'Bryant for more details. - Franklin T. Adams-Watters, May 16 2006
In a note on arXiv, "Disquisitiones Arithmeticae and online sequence A108345", I show that the upper density is at most 1/16. Furthermore computer evidence presented there suggests that the density is not 0, but 1/32. [From Paul Monsky (monsky(AT)brandeis.edu), Sep 24 2010]
LINKS
J. N. Cooper, D. Eichhorn and K. O'Bryant, Reciprocals of binary power series
Paul Monsky, Disquisitiones Arithmeticae and online sequence A108345 [From Paul Monsky (monsky(AT)brandeis.edu), Sep 24 2010]
FORMULA
The terms are the exponents in the expansion of 1/(Sum_{ n >= 0 } x^(n^2)) read mod 2. - N. J. A. Sloane, Dec 12 2007
MAPLE
N:= 500: # to get all terms <= N
S:= series(1/add(x^(n^2), n=0..floor(sqrt(N))), x, N+1) mod 2:
select(t -> coeff(S, x, t)=1, [$0..N]); # Robert Israel, Jun 01 2016
MATHEMATICA
Exponent[#, x]& /@ List @@ (Normal[2/(1+EllipticTheta[3, 0, x])+ O[x]^200] /. n_ x^k_ -> Mod[n, 2] x^k) (* Jean-François Alcover, Sep 17 2018 *)
CROSSREFS
Cf. A132229.
Sequence in context: A326917 A001857 A091532 * A073629 A125716 A171614
KEYWORD
nonn
AUTHOR
Ralf Stephan, Jul 01 2005
EXTENSIONS
More terms from Franklin T. Adams-Watters, May 16 2006
Changed comment "this set has zero density" to "this set is conjectured to have zero density". - Kevin O'Bryant, Jul 09 2010
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)