17

Part of 2014 Harvard-MIT Mathematics Tournament

Problems(1)

2014 Guts #17: Integer-Valued Function

Source:

f:\mathbb{N}\to\mathbb{N}

be a function satisfying the following conditions:
(a)

f(1)=1

f(a)\leq f(b)

a

b

are positive integers with

a\leq b

f(2a)=f(a)+1

for all positive integers

a

.
How many possible values can the

2014

(f(1),f(2),\ldots,f(2014))

functioninductionlogarithms