Problem Archive

Consider the infinite integer sequence S starting with:
$S = 1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 8, 4, 9, 1, 10, 11, 5, \dots$

Circle the first occurrence of each integer.
$S = \enclose{circle}1, 1, \enclose{circle}2, 1, \enclose{circle}3, 2, \enclose{circle}4, 1, \enclose{circle}5, 3, \enclose{circle}6, 2, \enclose{circle}7, \enclose{circle}8, 4, \enclose{circle}9, 1, \enclose{circle}{10}, \enclose{circle}{11}, 5, \dots$

The sequence is characterized by the following properties:

• The circled numbers are consecutive integers starting with $1$.
• Immediately preceding each non-circled numbers $a_i$, there are exactly $\lfloor \sqrt{a_i} \rfloor$ adjacent circled numbers, where $\lfloor\,\rfloor$ is the floor function.
• If we remove all circled numbers, the remaining numbers form a sequence identical to $S$, so $S$ is a fractal sequence.

Let $T(n)$ be the sum of the first $n$ elements of the sequence.
You are given $T(1) = 1$, $T(20) = 86$, $T(10^3) = 364089$ and $T(10^9) = 498676527978348241$.

Find $T(10^{18})$. Give the last $9$ digits of your answer.