MathDB

Set

p\ge 5

be a prime number and

n

be a natural number. Let

f

be a function

f: \mathbb{Z_{ \neq }}_0 \rightarrow \mathbb{ N }_0

satisfy the following conditions:
i) For all sequences of integers satisfy

a_i \not\in \{0, 1\}

, and

p

\not |

a_i-1

\forall

1 \le i \le p-2

,\\

\displaystyle \sum^{p-2}_{i=1}f(a_i)=f(a_1a_2 \cdots a_{p-2})

ii) For all coprime integers

a

and

b

a \equiv b \pmod p \Rightarrow f(a)=f(b)

iii) There exist

k \in \mathbb{Z}_{\neq 0}

that satisfy

f(k)=n

Prove that the number of such functions is

d(n)

, where

d(n)

denotes the number of divisors of

n

Problem Statement