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

1,1

Comments

An infinite sequence.

Links

Table of n, a(n) for n=1..58.

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

Cf. A004215, A018819, A363769.

Sequence in context: A118608 A176241 A075034 * A022110 A041808 A041810

Adjacent sequences: A363767 A363768 A363769 * A363771 A363772 A363773

Keyword

nonn,easy

Author

Maciej Ulas, Jun 21 2023

Status

approved