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A363769
Integers k such that the number of binary partitions of 2k is not a sum of three squares.
1
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
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
KEYWORD
nonn,easy
AUTHOR
Maciej Ulas, Jun 21 2023
STATUS
approved