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

Abstract

Let m be a positive integer and 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, then the natural density of n such that b_{m}(n) cannot be represented as a sum of three squares exists, and equals 1/12 for k=1, 2 and 1/6 for kge 3. In particular, for m=1 the equation 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 for i=0, 1 and k, s are non-negative integers, and where 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.