MathDB

Given natural numbers

a,b

and function

f: \mathbb{N} \to \mathbb{N}

such that for any natural number

n, f\left( n+a \right)

is divided by

f\left( {\left[ {\sqrt n } \right] + b} \right)

. Prove that for any natural

n

exist

n

pairwise distinct and pairwise relatively prime natural numbers

{{a}_{1}}

{{a}_{2}}

\ldots

{{a}_{n}}

such that the number

f\left( {{a}_{i+1}} \right)

is divided by

f\left( {{a}_{i}} \right)

for each

i=1,2, \dots ,n-1

.
(Here

[x]

is the integer part of number

x

, that is, the largest integer not exceeding

x

Problem Statement