MathDB

Let a sequence

(a_i)_{i=10}^{\infty}

be defined as follows:
[*]

a_{10}

is some positive integer, which can of course be written in base 10. [*] For

i \geq 10

a_i > 0

, let

b_i

be the positive integer whose base-

(i + 1)

representation is the same as

a_i

's base-

i

representation. Then let

a_{i + 1} = b_i - 1

. If

a_i = 0

a_{i + 1} = 0

.
For example, if

a_{10} = 11

, then

b_{10} = 11_{11} (= 12_{10})

;

a_{11} = 11_{11} - 1 = 10_{11} (= 11_{10})

;

b_{11} = 10_{12} (= 12_{10})

;

a_{12} = 11

.
Does there exist

a_{10}

such that

a_i

is strictly positive for all

i \geq 10

A5