Problem 3

For positive integer

k

, let

f(k)

be the number of integers in

\{k+1,\ldots,2k\}

whose binary expansion has exactly three

1

s. Prove every positive integer

m

occurs as

f(k)

, and determine all

m

for which exactly one

k

has

f(k)=m

Step 4 of 5: Find consecutive binary numbers with two ones

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In plain words

Find consecutive binary numbers with two ones

k-1=2^n+1,\quad k=2^n+2\quad(n\ge2)

Detailed analysis

If a positive integer and its successor both have two $1$ s, the lower integer must have one high $1$ and the final $1$ : $k-1=2^n+1$ . The condition that $k$ also has two $1$ s requires $n\ge2$ , giving $k=2^n+2$ .