25 lines
995 B
Markdown
25 lines
995 B
Markdown
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A natural number, N, that can be written as the sum and product of a given set of
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at least two natural numbers, `{a1, a2, ... , ak}` is called a product-sum number:
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`N = a1 + a2 + ... + ak = a1 * a2 * ... * ak`.
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For example, 6 = 1 + 2 + 3 = 1 × 2 × 3.
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For a given set of size, k, we shall call
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the smallest N with this property a minimal product-sum number.
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The minimal product-sum numbers for sets of size, `k = 2, 3, 4, 5`, and `6` are as follows.
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~~~
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k=2: 4 = 2 * 2 = 2 + 2
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k=3: 6 = 1 * 2 * 3 = 1 + 2 + 3
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k=4: 8 = 1 * 1 * 2 * 4 = 1 + 1 + 2 + 4
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k=5: 8 = 1 * 1 * 2 * 2 * 2 = 1 + 1 + 2 + 2 + 2
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k=6: 12 = 1 * 1 * 1 * 1 * 2 * 6 = 1 + 1 + 1 + 1 + 2 + 6
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~~~
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Hence for `2<=k<=6`, the sum of all the minimal product-sum numbers is `4+6+8+12 = 30`;
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note that 8 is only counted once in the sum.
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In fact, as the complete set of minimal product-sum numbers for `2<=k<=12`
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is `{4, 6, 8, 12, 15, 16}`, the sum is `61`.
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What is the sum of all the minimal product-sum numbers for `2<=k<=12000`?
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