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18004Re: additive combinations all prime?

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  • Patrick Capelle
    May 7, 2006
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      --- "Mark Underwood" <mark.underwood@...> wrote:
      >
      > Well surprise of surprises:
      >
      > The six numbers (5,30,33,42,60,63) yield primes for all 32 additive
      > combinations of the six numbers.(!) I thought it would have been
      > waaay higher.
      >
      > So we have as first cases:
      >
      > (1,4) has all 2 additive combinations yielding 2 primes.
      > (1,4,8) has all 4 additive combos yielding 4 primes.
      > (3,5,8,13) has all 8 additive combos yielding 8 primes.
      > (3,10,12,15,27) has all 16 additive combos yielding 16 primes.
      > (5,30,33,42,60,63) has all 32 additive combos yielding 32 primes.
      >
      > Interesting that thus far the greatest number in each set does not
      > quite exceed 2^n. Surely that can't hold up for long...or can it?
      >
      > Wonders never cease!
      > Mark
      ----------------------------------------------------------------------

      Something interesting with the product of the numbers ?

      (1,4)--> 4 = 2^2
      (1,4,8)--> 32 = 2^5
      (3,5,8,13)--> 1560 = 2^3 * 3 * 5 * 13
      (3,10,12,15,27)--> 145800 = 2^3 * 3^6 * 5^2
      (5,30,33,42,60,63)--> 785862000 = 2^4 * 3^6 * 5^3 * 7^2 * 11

      In each case the product of the n numbers gives a number whose number
      of different prime factors (in the factorization) is smaller or equal
      to n.
      Is it always the case ?

      Patrick Capelle.
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