MathDB

For a positive integer

k

, define the

k

-pop of a positive integer

n

as the infinite sequence of integers

a_1, a_2, ...

such that

a_1 = n

and

a_{i+1}= \left\lfloor \frac{a_i}{k} \right\rfloor , i = 1, 2, ..

where

\lfloor x\rfloor

denotes the greatest integer less than or equal to

x

. Furthermore, define a positive integer

m

to be

k

-pop avoiding if

k

does not divide any nonzero term in the

k

-pop of

m

. For example,

14

is 3-pop avoiding because

3

does not divide any nonzero term in the

3

-pop of

14

, which is

14, 4, 1, 0, 0, ....

Suppose that the number of positive integers less than

13^{2018}

which are

13

-pop avoiding is equal to N. What is the remainder when

N

is divided by

1000

7