30 lines
1.1 KiB
Plaintext
30 lines
1.1 KiB
Plaintext
;Definition
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A positive integer '''n''' is an arithmetic number if the average of its positive divisors is also an integer.
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Clearly all odd primes '''p''' must be arithmetic numbers because their only divisors are '''1''' and '''p''' whose sum is even and hence their average must be an integer. However, the prime number '''2''' is not an arithmetic number because the average of its divisors is 1.5.
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;Example
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30 is an arithmetic number because its 7 divisors are: [1, 2, 3, 5, 6, 10, 15, 30], their sum is 72 and average 9 which is an integer.
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;Task
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Calculate and show here:
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1. The first 100 arithmetic numbers.
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2. The '''x'''th arithmetic number where '''x''' = 1,000 and '''x''' = 10,000.
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3. How many of the first '''x''' arithmetic numbers are composite.
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Note that, technically, the arithmetic number '''1''' is neither prime nor composite.
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;Stretch
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Carry out the same exercise in 2. and 3. above for '''x''' = 100,000 and '''x''' = 1,000,000.
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;References
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* [[wp:Arithmetic_number|Wikipedia: Arithmetic number]]
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* [[oeis:A003601|OEIS:A003601 - Numbers n such that the average of the divisors of n is an integer]]
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<br><br>
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