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3671. Sum of Beautiful Subsequences

Description

You are given an integer array nums of length n.

For every positive integer g, we define the beauty of g as the product of g and the number of strictly increasing subsequences of nums whose greatest common divisor (GCD) is exactly g.

Return the sum of beauty values for all positive integers g.

Since the answer could be very large, return it modulo 109 + 7.

 

Example 1:

Input: nums = [1,2,3]

Output: 10

Explanation:

All strictly increasing subsequences and their GCDs are:

Subsequence GCD
[1] 1
[2] 2
[3] 3
[1,2] 1
[1,3] 1
[2,3] 1
[1,2,3] 1

Calculating beauty for each GCD:

GCD Count of subsequences Beauty (GCD × Count)
1 5 1 × 5 = 5
2 1 2 × 1 = 2
3 1 3 × 1 = 3

Total beauty is 5 + 2 + 3 = 10.

Example 2:

Input: nums = [4,6]

Output: 12

Explanation:

All strictly increasing subsequences and their GCDs are:

Subsequence GCD
[4] 4
[6] 6
[4,6] 2

Calculating beauty for each GCD:

GCD Count of subsequences Beauty (GCD × Count)
2 1 2 × 1 = 2
4 1 4 × 1 = 4
6 1 6 × 1 = 6

Total beauty is 2 + 4 + 6 = 12.

 

Constraints:

  • 1 <= n == nums.length <= 104
  • 1 <= nums[i] <= 7 * 104

Solutions

Solution 1

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1


1


1

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