Minimum Coin Flip Operations

You are given n coins arranged in a row. Initially, only the first coin is facing down while the remaining coins are facing up. In each operation, you may choose any positive integer \(i\) and flip (i.e. change from up to down or from down to up) every coin at positions \(j\) that satisfy \(j \bmod i = 0\). Your task is to determine the minimum number of operations required so that all coins are facing up.

Explanation:

Let the coins be numbered from \(1\) to \(n\). Define an operation on an integer \(i\) (with \(1 \le i \le n\)) which flips all coins at positions that are multiples of \(i\). Denote by \(x_i\) whether you perform an operation with \(i\) (1 if performed, 0 otherwise). Note that coin \(j\) is flipped for every \(i\) dividing \(j\), so its flip count is \(\sum_{i\mid j} x_i\).

The initial state is as follows:

• Coin 1: down (represented as 0)
• Coins 2 to \(n\): up (represented as 1)

After applying the chosen operations, the final state of coin \(j\) will be:

[ \text{final state of coin } j = \text{initial state} + \sum_{i \mid j} x_i \pmod{2}. ]

We want every coin to be up (i.e. 1). Thus, for coin 1 (initially 0) we need:

[ x_1 \equiv 1 \pmod{2}, ]

and for each coin \(j \ge 2\) (initially 1) we need:

[ 1 + \sum_{\substack{i \mid j}} x_i \equiv 1 \pmod{2} \quad \Longrightarrow \quad \sum_{\substack{i \mid j}} x_i \equiv 0 \pmod{2}. ]

Since coin 1 is flipped in all operations (because \(1 \bmod i = 0\) for every \(i\) when \(j=1\)), we must have \(x_1=1\). Then, for any coin \(j \ge 2\), notice that 1 is always a divisor, so the condition becomes:

[ 1 + \sum_{\substack{i \mid j,\ i>1}} x_i \equiv 0 \pmod{2} \quad \Longrightarrow \quad \sum_{\substack{i \mid j,\ i>1}} x_i \equiv 1 \pmod{2}. ]

This system of equations has a unique solution if we decide to minimize the total number of operations (i.e. the sum of \(x_i\) values) by setting:

[ x_1 = 1, \quad \text{and for } j \ge 2, \quad x_j = \left(1 + \sum_{\substack{i \mid j,\ 1<i<j}} x_i \right) \bmod 2. ]

The answer is then the total number of operations performed, i.e.,

[ \text{Answer} = \sum_{j=1}^{n} x_j. ]

Example:

• If \(n = 4\):
  • Coin 1: \(x_1 = 1\).
  • Coin 2: No divisors between 2 and 1, so \(x_2 = 1\) (to cancel the effect of coin 1, since \(1 + x_2 \equiv 0 \pmod{2}\)).
  • Coin 3: Similarly, \(x_3 = 1\).
  • Coin 4: Divisors (other than 1 and itself) include 2. Thus, \(x_4 = (1 + x_2) \bmod 2 = 0\).
  The total number of operations is \(1 + 1 + 1 + 0 = 3\).

inputFormat

The input consists of a single integer \(n\) (\(1 \le n \le 10^5\) or as specified by the problem constraints) representing the number of coins.

outputFormat

Output a single integer, the minimum number of operations required to make all coins face up.

sample

1

1