MathDB

2014 Guts #17: Integer-Valued Function

Source:

April 21, 2014

functioninductionlogarithms

Problem Statement

Let

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

be a function satisfying the following conditions:
(a)

f(1)=1

. (b)

f(a)\leq f(b)

whenever

a

and

b

are positive integers with

a\leq b

. (c)

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

for all positive integers

a

.
How many possible values can the

2014

-tuple

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

take?