MathDB

For a given odd prime number

p

, define

f(n)

the remainder of

d

divided by

p

, where

d

is the biggest divisor of

n

which is not a multiple of

p

. For example when

p=5

f(6)=1, f(35)=2, f(75)=3

. Define the sequence

a_1, a_2, \ldots, a_n, \ldots

of integers as the followings:
[*]

a_1=1

[*]

a_{n+1}=a_n+(-1)^{f(n)+1}

for all positive integers

n

.
Determine all integers

m

, such that there exist infinitely many positive integers

k

such that

m=a_k

3