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.

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

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

Robert Israel, Table of n, a(n) for n = 1..10000

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

Adjacent sequences: A108342 A108343 A108344 * A108346 A108347 A108348

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