A363770 Integers k such that the number of binary partitions of k is not a sum of three squares.
20, 21, 36, 37, 68, 69, 80, 81, 116, 117, 132, 133, 144, 145, 180, 181, 212, 213, 228, 229, 260, 261, 272, 273, 308, 309, 320, 321, 340, 341, 356, 357, 404, 405, 420, 421, 452, 453, 464, 465, 500, 501, 516, 517, 528, 529, 564, 565, 576, 577, 596, 597, 612, 613, 660, 661, 676, 677
Offset: 1
Examples
a(1)=20 because b(20)=60 is not a sum of three squares and for i=1, ..., 19, the numbers b(i), i=1,...,19 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.
Programs
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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]]]]; B := {}; Do[ If[Mod[bin[n]/4^IntegerExponent[bin[n], 4], 8] == 7, AppendTo[B, n]], {n, 1000}]; B
Formula
Each term is equal to 2*b(m) or 2*b(m)+1 for some m, where b(m) = A363769(m).
Comments