Cards

Lukas loves card games and has invented a new way to shuffle a deck.

There are $N$ cards, numbered $1$ to $N$, stacked in the first pile in some order. A second pile starts out empty. You perform exactly $N$ operations. Let $p_k$ be the $k$-th prime number ($p_1 = 2$, $p_2 = 3$, $p_3 = 5$, $\dots$). In the $k$-th operation you:

1. move $p_k - 1$ cards, one at a time, from the top of the first pile to its bottom, then
2. take the card now on top of the first pile and place it on top of the second pile.

For example, suppose the first pile holds $1, 2, 3, 4, 5, 6, 7$ from top to bottom. Performing an operation with $p = 5$ first moves the top $4$ cards to the bottom, giving $5, 6, 7, 1, 2, 3, 4$; then the top card $5$ is removed and placed on the second pile, leaving $6, 7, 1, 2, 3, 4$.

Arrange the cards in the first pile so that after all $N$ operations the second pile reads, from top to bottom, $N, N-1, \dots, 2, 1$ (that is, card $1$ ends up at the very bottom of the second pile).

A single integer $N$ ($2 \le N \le 100\,000$) — the number of cards in the first pile.

Output $N$ lines. The $i$-th line contains $a_i$, the number of the card at position $i$ in the first pile, where position $1$ is the top card and position $N$ is the bottom card.

The first four primes are $2, 3, 5, 7$. Starting from the arrangement $2, 1, 3, 4$ (top to bottom), the first pile evolves as follows through the four operations:

$(2, 1, 3, 4) \to (3, 4, 2) \to (3, 4) \to (4) \to (\,)$

Operation $1$ ($p_1 = 2$): move card $2$ to the bottom, then take card $1$. Operation $2$ ($p_2 = 3$): after two moves the top card is $2$, which is taken. Operations $3$ and $4$ take cards $3$ and $4$. The second pile, filled from bottom to top, becomes $1, 2, 3, 4$ — that is $4, 3, 2, 1$ read from the top.