Problems & Puzzles: Puzzles

 Puzzle 600. Follow up to Puzzle 599 Fred Schneider sent the following follow up to puzzle 599. Q1 are there any weird numbers X = W * N where W is a weird number and N is NOT a prime > s(W) and whose smallest factor is < s(W).. Q2 In particular, are there any numbers where N is a prime and W < N < s(W).   Q3 If you can't find examples for either, can you prove or disprove their existence? _________ *s(W) is the sum of all of W's divisors. - Show quoted text -

Contribution came from Emmanuel Vantieghem.

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Emmanuel wrote:

Q1 : there are such numbers. The smallest example is  W = 836  and  N = 421 (it is prime, but not a prime greater than s(W) = 844 ; I guessed N is not required to be composite).

Q2 : take  W = 836 and N = 839.

I found many weird numbers  w  for which the smallest prime  p  that makes  w p  is weird is much smaller than  w.  For instance  w = 7192, where  p = 31.  In general, it is possible to prove a stronger result than that of puzzle 599: if  w  is a weird number and  p  a prime bigger than the sum of all the divisors of  w, then  w p^n  is weird for all  n > 0. (could it be a challenge for some puzzlers to recover my proof ?)

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