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<p>You are given an integer array <code>nums</code> of length <code>n</code>.</p>
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<p>For every <strong>positive</strong> integer <code>g</code>, we define the <strong>beauty</strong> of <code>g</code> as the <strong>product</strong> of <code>g</code> and the number of <strong>strictly increasing</strong> <strong><span data-keyword="subsequence-array-nonempty">subsequences</span></strong> of <code>nums</code> whose greatest common divisor (GCD) is exactly <code>g</code>.</p>
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<p>Return the <strong>sum</strong> of <strong>beauty</strong> values for all positive integers <code>g</code>.</p>
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<p>Since the answer could be very large, return it modulo <code>10<sup>9</sup> + 7</code>.</p>
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<p> </p>
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<p><strong class="example">Example 1:</strong></p>
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<div class="example-block">
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<p><strong>Input:</strong> <span class="example-io">nums = [1,2,3]</span></p>
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<p><strong>Output:</strong> <span class="example-io">10</span></p>
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<p><strong>Explanation:</strong></p>
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<p>All strictly increasing subsequences and their GCDs are:</p>
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<table style="border: 1px solid black;">
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<thead>
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<tr>
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<th style="border: 1px solid black;">Subsequence</th>
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<th style="border: 1px solid black;">GCD</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td style="border: 1px solid black;">[1]</td>
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<td style="border: 1px solid black;">1</td>
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</tr>
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<tr>
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<td style="border: 1px solid black;">[2]</td>
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<td style="border: 1px solid black;">2</td>
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</tr>
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<tr>
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<td style="border: 1px solid black;">[3]</td>
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<td style="border: 1px solid black;">3</td>
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</tr>
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<tr>
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<td style="border: 1px solid black;">[1,2]</td>
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<td style="border: 1px solid black;">1</td>
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</tr>
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<tr>
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<td style="border: 1px solid black;">[1,3]</td>
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<td style="border: 1px solid black;">1</td>
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</tr>
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<tr>
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<td style="border: 1px solid black;">[2,3]</td>
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<td style="border: 1px solid black;">1</td>
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</tr>
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<tr>
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<td style="border: 1px solid black;">[1,2,3]</td>
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<td style="border: 1px solid black;">1</td>
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</tr>
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</tbody>
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</table>
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<p>Calculating beauty for each GCD:</p>
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<table style="border: 1px solid black;">
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<thead>
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<tr>
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<th style="border: 1px solid black;">GCD</th>
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<th style="border: 1px solid black;">Count of subsequences</th>
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<th style="border: 1px solid black;">Beauty (GCD × Count)</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td style="border: 1px solid black;">1</td>
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<td style="border: 1px solid black;">5</td>
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<td style="border: 1px solid black;">1 × 5 = 5</td>
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</tr>
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<tr>
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<td style="border: 1px solid black;">2</td>
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<td style="border: 1px solid black;">1</td>
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<td style="border: 1px solid black;">2 × 1 = 2</td>
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</tr>
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<tr>
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<td style="border: 1px solid black;">3</td>
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<td style="border: 1px solid black;">1</td>
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<td style="border: 1px solid black;">3 × 1 = 3</td>
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</tr>
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</tbody>
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</table>
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<p>Total beauty is <code>5 + 2 + 3 = 10</code>.</p>
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</div>
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<p><strong class="example">Example 2:</strong></p>
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<div class="example-block">
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<p><strong>Input:</strong> <span class="example-io">nums = [4,6]</span></p>
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<p><strong>Output:</strong> <span class="example-io">12</span></p>
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<p><strong>Explanation:</strong></p>
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<p>All strictly increasing subsequences and their GCDs are:</p>
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<table style="border: 1px solid black;">
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<thead>
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<tr>
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<th style="border: 1px solid black;">Subsequence</th>
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<th style="border: 1px solid black;">GCD</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td style="border: 1px solid black;">[4]</td>
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<td style="border: 1px solid black;">4</td>
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</tr>
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<tr>
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<td style="border: 1px solid black;">[6]</td>
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<td style="border: 1px solid black;">6</td>
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</tr>
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<tr>
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<td style="border: 1px solid black;">[4,6]</td>
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<td style="border: 1px solid black;">2</td>
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</tr>
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</tbody>
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</table>
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<p>Calculating beauty for each GCD:</p>
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<table style="border: 1px solid black;">
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<thead>
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<tr>
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<th style="border: 1px solid black;">GCD</th>
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<th style="border: 1px solid black;">Count of subsequences</th>
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<th style="border: 1px solid black;">Beauty (GCD × Count)</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td style="border: 1px solid black;">2</td>
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<td style="border: 1px solid black;">1</td>
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<td style="border: 1px solid black;">2 × 1 = 2</td>
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</tr>
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<tr>
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<td style="border: 1px solid black;">4</td>
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<td style="border: 1px solid black;">1</td>
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<td style="border: 1px solid black;">4 × 1 = 4</td>
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</tr>
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<tr>
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<td style="border: 1px solid black;">6</td>
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<td style="border: 1px solid black;">1</td>
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<td style="border: 1px solid black;">6 × 1 = 6</td>
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</tr>
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</tbody>
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</table>
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<p>Total beauty is <code>2 + 4 + 6 = 12</code>.</p>
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</div>
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<p> </p>
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<p><strong>Constraints:</strong></p>
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<ul>
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<li><code>1 <= n == nums.length <= 10<sup>4</sup></code></li>
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<li><code>1 <= nums[i] <= 7 * 10<sup>4</sup></code></li>
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</ul>
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