MathDB

Let

N

be a positive integer. Consider the sequence

a_1, a_2, ..., a_N

of positive integers, none of which is a multiple of

2^{N+1}

. For

n \ge N +1

, the number

a_n

is defined as follows: choose

k

to be the number among

1, 2, ..., n - 1

for which the remainder obtained when

a_k

is divided by

2^n

is the smallest, and define

a_n = 2a_k

(if there are more than one such

k

, choose the largest such

k

). Prove that there exist

M

for which

a_n = a_M

holds for every

n \ge M

Problem Statement