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 Sequence starting with 30 (Posted on 2016-07-05)
The prime factors of 30 are {2,3,5}
The sum of the reciprocals is 1/2+1/3+1/5=31/30
The product of the reciprocals is 1/2*1/3*1/5=1/30
The difference of the two is 31/30-1/30=1

There aren't many numbers where the final value of this procedure is a whole number.

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Show that they cannot be semi-prime.
Show they must be square-free.

 No Solution Yet Submitted by Jer No Rating

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 Square-free (proof) Comment 3 of 3 |
Well, let's prove that the prime factors cannot be a, a, b and c.

If 1/a + 1/a + 1/b + 1/c -1/a*a*b*c = 1

Then  (2abc + a*a*c + a*a*b - 1) = a*a*b*c

the RHS is a multiple of a, but the LHS is not.

This is a contradiction, so the prime factors cannot be a, a, b and c.

Similarly, they cannot be a, a, and anything else.

So the product must be square-free.

 Posted by Steve Herman on 2016-07-05 18:26:14

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