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Problem of the Week 917

Explosion

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A set A of (distinct) positive integers is "special above k" if every x in A such that x > k satisfies:

  1. x divides the product of all y in A with y < x
  2. x does not divide any y in A with y > x.

Example: {1, 2, 3, 6, 9} is special above 3 (this is the largest set that is special above 3) and {1, 2, 4, 8} is special above 4 but not special above 3.

Find as large a set as you can that is special above 4.
Find as large a set as you can that is special above 5.

Feel free to investigate farther... the shocking details of what is known will come next week.

Source: Harvey Friedman, Ohio State University
© Copyright 2000 Stan Wagon. Reproduced with permission.

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26 September 2000