Adjacent GCD

AtCoder

IDarc185_e

Time2000ms

Memory256MB

Difficulty—

Define the **score** of a sequence of positive integers $B = (B_1, B_2, \dots, B_k)$ as $\displaystyle \sum_{i=1}^{k-1} \gcd(B_i, B_{i+1})$. Given a sequence of positive integers $A = (A_1, A_2, \dots, A_N)$, solve the following problem for $m = 1, 2, \dots, N$. * There are $2^m - 1$ non-empty subsequences of the sequence $(A_1, A_2, \dots, A_m)$. Find the sum of the scores of all those subsequences, modulo $998244353$. Two subsequences are distinguished if they are taken from different positions in the sequence, even if they coincide as sequences. ## Constraints * $1 \leq N \leq 5 \times 10^5$ * $1 \leq A_i \leq 10^5$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $A_1$ $A_2$ $\dots$ $A_N$ [samples]

Samples

Input #1

3
9 6 4

Output #1

0
3
11

Consider the case $m = 3$. Here are the non-empty subsequences of $(A_1, A_2, A_3) = (9, 6, 4)$ and their scores.

*   $(9)$: Score is $0$.
*   $(6)$: Score is $0$.
*   $(4)$: Score is $0$.
*   $(9, 6)$: Score is $\gcd(9, 6) = 3$.
*   $(9, 4)$: Score is $\gcd(9, 4) = 1$.
*   $(6, 4)$: Score is $\gcd(6, 4) = 2$.
*   $(9, 6, 4)$: Score is $\gcd(9, 6) + \gcd(6, 4) = 3 + 2 = 5$.

Therefore, the answer for $m = 3$ is $0 + 0 + 0 + 3 + 1 + 2 + 5 = 11$.

Input #2

5
3 8 12 6 9

Output #2

Input #3

10
47718 21994 74148 76721 98917 73766 29598 59035 69293 29127

Output #3

API Response (JSON)

{
  "problem": {
    "name": "Adjacent GCD",
    "description": {
      "content": "Define the **score** of a sequence of positive integers $B = (B_1, B_2, \\dots, B_k)$ as $\\displaystyle \\sum_{i=1}^{k-1} \\gcd(B_i, B_{i+1})$.   Given a sequence of positive integers $A = (A_1, A_2, \\do",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc185_e"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Define the **score** of a sequence of positive integers $B = (B_1, B_2, \\dots, B_k)$ as $\\displaystyle \\sum_{i=1}^{k-1} \\gcd(B_i, B_{i+1})$.  \nGiven a sequence of positive integers $A = (A_1, A_2, \\do...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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