Values of Binary Partition Function Represented by a Sum of Three Squares

Abstract

Let m be a positive integer and $$b_{m}(n)$$ b m ( n ) be the number of partitions of a non-negative integer n with parts being powers of 2, where each part can take m colors. We show that if $$m=2^{k}-1$$ m = 2 k - 1 , then the natural density of n such that $$b_{m}(n)$$ b m ( n ) cannot be represented as a sum of three squares exists, and equals 1/12 for $$k=1, 2$$ k = 1 , 2 and 1/6 for $$k\ge 3$$ k ≥ 3 . In particular, for $$m=1$$ m = 1 the equation $$b_{1}(n)=x^2+y^2+z^2$$ b 1 ( n ) = x 2 + y 2 + z 2 has a solution in integers if and only if n is not of the form $$2^{2k+2}(8s+2t_{s}+3)+i$$ 2 2 k + 2 ( 8 s + 2 t s + 3 ) + i for $$i=0, 1$$ i = 0 , 1 and k, s are non-negative integers, and where $$t_{n}$$ t n is the nth term in the Prouhet–Thue–Morse sequence. A similar characterization is obtained for the solutions in n of the equation $$b_{2^k-1}(n)=x^2+y^2+z^2$$ b 2 k - 1 ( n ) = x 2 + y 2 + z 2 .