MathDB

n

-tuple

(a_1, a_2 . . . , a_n)

is occasionally periodic if there exist a non-negative integer

i

and a positive integer

p

satisfying

i + 2p \le n

and

a_{i+j} = a_{i+j+p}

for every

j = 1, 2, . . . , p

. Let

k

be a positive integer. Find the least positive integer

n

for which there exists an

n

-tuple

(a_1, a_2 . . . , a_n)

with elements from the set

\{1, 2, . . . , k\}

, which is not occasionally periodic but whose arbitrary extension

(a_1, a_2, . . . , a_n, a_{n+1})

is occasionally periodic for any

a_{n+1} \in \{1, 2, . . . , k\}

Problem Statement