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A363770
Integers k such that the number of binary partitions of k is not a sum of three squares.
0
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
1,1
COMMENTS
An infinite sequence.
LINKS
Bartosz Sobolewski and Maciej Ulas, Values of binary partition function represented by a sum of three squares, arXiv:2211.16622 [math.NT], 2023.
FORMULA
Each term is equal to 2*b(m) or 2*b(m)+1 for some m, where b(m) = A363769(m).
EXAMPLE
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.
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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Maciej Ulas, Jun 21 2023
STATUS
approved