MathDB

For positive integers

n,

the numbers

f(n)

are defined inductively as follows: f(1) \equal{} 1, and for every positive integer

n,

f(n\plus{}1) is the greatest integer

m

such that there is an arithmetic progression of positive integers a_1 < a_2 < \ldots < a_m \equal{} n for which f(a_1) \equal{} f(a_2) \equal{} \ldots \equal{} f(a_m). Prove that there are positive integers

a

and

b

such that f(an\plus{}b) \equal{} n\plus{}2 for every positive integer

n.

Problem Statement