MathDB

Consider the function

f_k:\mathbb{Z}^{+}\rightarrow\mathbb{Z}^{+}

satisfying

f_k(x)=x+k\varphi(x)

where

\varphi(x)

is Euler's totient function, that is, the number of positive integers up to

x

coprime to

x

. We define a sequence

a_1,a_2,...,a_{10}

with
[*]

a_1=c

, and [*]

a_n=f_k(a_{n-1}) \text{ }\forall \text{ } 2\le n\le 10

Is it possible to choose the initial value

c\ne 1

such that each term is a multiple of the previous, if (a)

k=2025

? (b)

k=2065

?
Proposed by chorn

Problem Statement