MathDB

Consecutive 2-Niven numbers

Source: Kürschak 2012, problem 2

July 6, 2014

modular arithmeticcombinatorics unsolvedcombinatorics

Problem Statement

Denote by

E(n)

the number of

1

's in the binary representation of a positive integer

n

. Call

n

interesting if

E(n)

divides

n

. Prove that (a) there cannot be five consecutive interesting numbers, and (b) there are infinitely many positive integers

n

such that

n

n+1

and

n+2

are each interesting.