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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.
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%I #21 Sep 17 2018 08:46:53

%S 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,

%T 55,59,61,63,71,72,73,79,81,83,87,89,91,97,98,101,103,107,109,113,115,

%U 117,121,127,128,137,139,149,151,153,157,159,162,167,171,173,181,183,191

%N 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.

%C 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

%C 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]

%H Robert Israel, <a href="/A108345/b108345.txt">Table of n, a(n) for n = 1..10000</a>

%H J. N. Cooper, D. Eichhorn and K. O'Bryant, <a href="http://arXiv.org/abs/math.NT/0506496">Reciprocals of binary power series</a>

%H Paul Monsky, <a href="http://arxiv.org/abs/1009.3985">Disquisitiones Arithmeticae and online sequence A108345</a> [From Paul Monsky (monsky(AT)brandeis.edu), Sep 24 2010]

%F 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

%p N:= 500: # to get all terms <= N

%p S:= series(1/add(x^(n^2),n=0..floor(sqrt(N))), x, N+1) mod 2:

%p select(t -> coeff(S,x,t)=1, [$0..N]); # _Robert Israel_, Jun 01 2016

%t 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 *)

%Y Cf. A132229.

%K nonn

%O 1,3

%A _Ralf Stephan_, Jul 01 2005

%E More terms from _Franklin T. Adams-Watters_, May 16 2006

%E Changed comment "this set has zero density" to "this set is conjectured to have zero density". - _Kevin O'Bryant_, Jul 09 2010