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March 26, 2017

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The sequence {ak}112 (base)k=1 satisfies a1=1 and an=1337+n/an−1, for all positive integers n. Let
S=⌊a10a13+a11a14+a12a15+⋯+a109a112⌋.
Find the remainder when S is divided by 1000.

Details and assumptions
The function ⌊x⌋:R→Z refers to the greatest integer smaller than or equal to x. For example ⌊2.3⌋=2 and ⌊−5⌋=−5.

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