MathDB

Given two positive integers

x,y

, we define

z=x\,\oplus\,y

to be the bitwise XOR sum of

x

and

y

; that is,

z

has a

1

in its binary representation at exactly the place values where

x,y

have differing binary representations. It is known that

\oplus

is both associative and commutative. For example,

20 \oplus 18 = 10100_2 \oplus 10010_2 = 110_2 = 6

. Given a set

S=\{a_1, a_2, \dots, a_n\}

of positive integers, we let

f(S) = a_1 \oplus a_2 \oplus a_3\oplus \dots \oplus a_n

. We also let

g(S)

be the number of divisors of

f(S)

which are at most

2018

but greater than or equal to the largest element in

S

(if

S

is empty then let

g(S)=2018

). Compute the number of

1

s in the binary representation of

\displaystyle\sum_{S\subseteq \{1,2,\dots, 2018\}} g(S)

.
Proposed by Brandon Wang and Vincent Huang

Problem Statement