Partitions of natural numbers with the same representation functions

Abstract

For a set A of nonnegative integers the representation functions R 2 ( A , n ) , R 3 ( A , n ) are defined as the number of solutions of the equation n = a + a ′ , a , a ′ ∈ A with a < a ′ , a ⩽ a ′ , respectively. Let D ( 0 ) = 0 and let D ( a ) denote the number of ones in the binary representation of a. Let A 0 be the set of all nonnegative integers a with even D ( a ) and A 1 be the set of all nonnegative integers a with odd D ( a ) . In this paper we show that (a) if R 2 ( A , n ) = R 2 ( N ∖ A , n ) for all n ⩾ 2 N − 1 , then R 2 ( A , n ) = R 2 ( N ∖ A , n ) ⩾ 1 for all n ⩾ 12 N 2 − 10 N − 2 except for A = A 0 or A = A 1 ; (b) if R 3 ( A , n ) = R 3 ( N ∖ A , n ) for all n ⩾ 2 N − 1 , then R 3 ( A , n ) = R 3 ( N ∖ A , n ) ⩾ 1 for all n ⩾ 12 N 2 + 2 N . Several problems are posed in this paper.