cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

This page as a plain text file.
%I A363769 #38 Apr 13 2025 07:11:41
%S A363769 10,18,34,40,58,66,72,90,106,114,130,136,154,160,170,178,202,210,226,
%T A363769 232,250,258,264,282,288,298,306,330,338,354,360,378,394,402,418,424,
%U A363769 442,450,456,474,490,498,514,520,538,544,554,562,586,594,610,616,634,640,650,658,674,680,698
%N A363769 Integers k such that the number of binary partitions of 2k is not a sum of three squares.
%C A363769 An infinite sequence.
%H A363769 Bartosz Sobolewski and Maciej Ulas, <a href="https://arxiv.org/abs/2211.16622">Values of binary partition function represented by a sum of three squares</a>, arXiv:2211.16622 [math.NT], 2023.
%F A363769 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).
%e A363769 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.
%t A363769 bin[n_] :=
%t A363769  bin[n] =
%t A363769   If[n == 0, 1,
%t A363769    If[Mod[n, 2] == 0, bin[n - 1] + bin[n/2],
%t A363769     If[Mod[n, 2] == 1, bin[n - 1]]]];
%t A363769 A := {}; Do[
%t A363769  If[Mod[bin[2 n]/4^IntegerExponent[bin[2 n], 4], 8] == 7,
%t A363769   AppendTo[A, n]], {n, 1000}];
%t A363769 A
%Y A363769 Cf. A018819, A353219, A010060 (0 -> 1 & 1 -> -1).
%K A363769 nonn,easy
%O A363769 1,1
%A A363769 _Maciej Ulas_, Jun 21 2023