MathDB

Define the sequence

(a_n)

a_1 = 1

a_{2n} = a_n

and

a_{2n+1} = a_n + 1

for all

n\geq 1

.
a) Find all positive integers

n

such that

a_{kn} = a_n

for all integers

1 \leq k \leq n

. b) Prove that there exist infinitely many positive integers

m

such that

a_{km} \geq a_m

for all positive integers

k

1