A363769 Integers k such that the number of binary partitions of 2k is not a sum of three squares.

10, 18, 34, 40, 58, 66, 72, 90, 106, 114, 130, 136, 154, 160, 170, 178, 202, 210, 226, 232, 250, 258, 264, 282, 288, 298, 306, 330, 338, 354, 360, 378, 394, 402, 418, 424, 442, 450, 456, 474, 490, 498, 514, 520, 538, 544, 554, 562, 586, 594, 610, 616, 634, 640, 650, 658, 674, 680, 698

Offset: 1

Author

Maciej Ulas, Jun 21 2023

Keywords

• nonn
• easy

Comments

An infinite sequence.

Examples

			a(1)=10 because each b(20)=60 is not a sum of three squares and for i=1, ..., 9, the numbers b(2)=2, b(4)=4, b(6)=6, b(8)=10, b(10)=14, b(12)=20, b(14)=26, b(16)=36, b(18)=46 are sums of three squares, where b(i) is the number of binary partitions of n.
		

Links

• Bartosz Sobolewski and Maciej Ulas, Values of binary partition function represented by a sum of three squares, arXiv:2211.16622 [math.NT], 2023.

Crossrefs

Cf. A018819, A353219, A010060 (0 -> 1 & 1 -> -1).

Programs

Mathematica

bin[n_] :=
 bin[n] =
  If[n == 0, 1,
   If[Mod[n, 2] == 0, bin[n - 1] + bin[n/2],
    If[Mod[n, 2] == 1, bin[n - 1]]]];
A := {}; Do[
 If[Mod[bin[2 n]/4^IntegerExponent[bin[2 n], 4], 8] == 7,
  AppendTo[A, n]], {n, 1000}];
A

Formula

Numbers of the form {2^(2*k+1)*(8*r+2*t_{r}+3): k, r nonnegative integers} and t_{r} is r-th term of the Prouhet-Thue-Morse sequence on the alphabet {-1, +1}, i.e., t_{r} = (-1)^{s_{2}(r)}, where s_{2}(r) is the sum of binary digits of r. We have t_{r} = (-1)^A010060(r).