Partitions of natural numbers with the same weighted representation functions

TextFor any given two positive integers k1 and k2, and any set A of nonnegative integers, let rk1,k2(A,n) denote the number of solutions of the equation n=k1a1+k2a2 with a1,a2∈A. In this paper, we determine all pairs k1,k2 of positive integers for which there exists a set A⊆N such that rk1,k2(A,n)=rk1,k2(N∖A,n) for all n⩾n0. We also pose several problems for further research.VideoFor a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=EnezEsJl0OY.