A363770 - OEIS

%I #30 Aug 02 2023 12:05:38

%S 20,21,36,37,68,69,80,81,116,117,132,133,144,145,180,181,212,213,228,

%T 229,260,261,272,273,308,309,320,321,340,341,356,357,404,405,420,421,

%U 452,453,464,465,500,501,516,517,528,529,564,565,576,577,596,597,612,613,660,661,676,677

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

%C An infinite sequence.

%H 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 Each term is equal to 2*b(m) or 2*b(m)+1 for some m, where b(m) = A363769(m).

%e 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.

%t bin[n_] :=

%t  bin[n] =

%t   If[n == 0, 1,

%t    If[Mod[n, 2] == 0, bin[n - 1] + bin[n/2],

%t     If[Mod[n, 2] == 1, bin[n - 1]]]];

%t B := {}; Do[

%t  If[Mod[bin[n]/4^IntegerExponent[bin[n], 4], 8] == 7,

%t   AppendTo[B, n]], {n, 1000}];

%t B

%Y Cf. A004215, A018819, A363769.

%K nonn,easy

%O 1,1

%A _Maciej Ulas_, Jun 21 2023