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

1,1

Comments

An infinite sequence.

Links

Table of n, a(n) for n=1..59.

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

Formula

Numbers of the form 2^(2k+1)*(8r+2t_{r}+3): k, r positive 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(n).

Example

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.

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

Crossrefs

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

Sequence in context: A233695 A014006 A090995 * A153360 A189323 A064485

Adjacent sequences: A363766 A363767 A363768 * A363770 A363771 A363772

Keyword

nonn,easy

Author

Maciej Ulas, Jun 21 2023

Status

approved