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Fun with (eeek!) composites

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  • Mark Underwood
    As they relate to primes, of course :) Fun with successive numbers: 18,164,161 is a prime times 1 18,164,162 is a prime times 2 18,164,163 is a prime times 3
    Message 1 of 3 , Apr 4 11:44 AM
      As they relate to primes, of course :)

      Fun with successive numbers:


      18,164,161 is a prime times 1
      18,164,162 is a prime times 2
      18,164,163 is a prime times 3
      18,164,164 is a prime times 4
      18,164,165 is a prime times 5
      18,164,166 is a prime times 6
      18,164,167 is a prime times 7.

      *******

      34,415,168 is divisible by 2
      34,415,169 is divisible by 3
      34,415,170 is divisible by 5
      34,415,171 is divisible by 7
      34,415,172 is divisible by 11
      34,415,173 is divisible by 13
      34,415,174 is divisible by 17
      34,415,175 is divisible by 19
      34,415,176 is divisible by 23.


      ************

      12,252,242 is divisible by 2
      12,252,243 is divisible by 3
      12,252,244 is divisible by 4
      12,252,245 is divisible by 5
      12,252,246 is divisible by 6
      12,252,247 is divisible by 7
      12,252,248 is divisible by 8
      12,252,249 is divisible by 9
      12,252,250 is divisible by 10
      12,252,251 is divisible by 11
      12,252,252 is divisible by 12
      12,252,253 is divisible by 13
      12,252,254 is divisible by 14
      12,252,255 is divisible by 15
      12,252,256 is divisible by 16
      12,252,257 is divisible by 17
      12,252,258 is divisible by 18.

      **********

      Mark
    • Jens Kruse Andersen
      ... 2,918,756,139,031,688,155,200 + k is a prime times k, for k = 1..14 See http://www.primepuzzles.net/puzzles/puzz_181.htm 5,516,280 + k is the smallest
      Message 2 of 3 , Apr 4 4:13 PM
        Mark Underwood wrote:

        > Fun with successive numbers:
        >
        >
        > 18,164,161 is a prime times 1
        ...
        > 18,164,167 is a prime times 7.

        2,918,756,139,031,688,155,200 + k is a prime times k, for k = 1..14
        See http://www.primepuzzles.net/puzzles/puzz_181.htm
        5,516,280 + k is the smallest solution for k = 1..7

        > 34,415,168 is divisible by 2
        > 34,415,169 is divisible by 3
        > 34,415,170 is divisible by 5
        > 34,415,171 is divisible by 7
        > 34,415,172 is divisible by 11
        > 34,415,173 is divisible by 13
        > 34,415,174 is divisible by 17
        > 34,415,175 is divisible by 19
        > 34,415,176 is divisible by 23.

        This is what CRT (Chinese Remainder Theorem) is for. A bigint program can easily
        reach primes in the millions but I will spare you solutions with millions of
        digits.

        > 12,252,242 is divisible by 2
        ...
        > 12,252,258 is divisible by 18.

        Because lcm(2,...,18) = 2^4 * 3^2 * 5 * 7 * 11 * 13 * 17 = 12,252,240
        (lcm = least common multiple)


        My contribution to fun with successive numbers is the smallest case of 10
        numbers where the nth has n prime factors:

        3931520917431241 = 3931520917431241
        3931520917431242 = 2 * 1965760458715621
        3931520917431243 = 3 * 221477 * 5917124453
        3931520917431244 = 2 * 2 * 23 * 42733923015557
        3931520917431245 = 5 * 17 * 199 * 5557 * 41826179
        3931520917431246 = 2 * 3 * 29 * 83 * 261799 * 1039837
        3931520917431247 = 7 * 7 * 7 * 19 * 43 * 557 * 25187741
        3931520917431248 = 2 * 2 * 2 * 2 * 31 * 59 * 167 * 804471071
        3931520917431249 = 3 * 3 * 3 * 3 * 3 * 41 * 53 * 1721 * 4326271
        3931520917431250 = 2 * 5 * 5 * 5 * 5 * 5 * 11 * 13 * 13 * 338377271

        The nth number is divisible by n above, but not in the 2nd smallest case
        starting at 5,818,684,827,160,441.

        See http://www.research.att.com/projects/OEIS?Anum=A072875

        --
        Jens Kruse Andersen
      • Mark Underwood
        Hi Jens The (N+k)/k prime puzzle. If I ve read it before I don t remember it! Perhaps it was posted on this site a year ago or so when I was visiting, and it
        Message 3 of 3 , Apr 4 6:30 PM
          Hi Jens

          The (N+k)/k prime puzzle. If I've read it before I don't remember it!
          Perhaps it was posted on this site a year ago or so when I was
          visiting, and it lodged in my subconscious :)


          I couldn't believe I sent the 12,252,240 + n is divisible by n thing.
          I realized a minute after I sent it what I had done, but it was too
          late!


          The Nth number has N prime factors puzzle is brilliant.
          That the Nth number is also divisible by N is not too improbable, on
          inspection. The 2,3,6 8th and 9th numbers *must* be divisible by N.
          Getting both the 5th and 10th numbers divisible by 5 and 10 is about
          a four in five chance, assuming there is not some unknown principle
          which might make it more certain or even guaranteed. Getting the
          seventh number divisible by 7 is about fifty fifty, again assuming
          there is not some unknown princple stacking the deck further.


          Mark










          --- In primenumbers@yahoogroups.com, "Jens Kruse Andersen"
          <jens.k.a@g...> wrote:
          > Mark Underwood wrote:
          >
          > > Fun with successive numbers:
          > >
          > >
          > > 18,164,161 is a prime times 1
          > ...
          > > 18,164,167 is a prime times 7.
          >
          > 2,918,756,139,031,688,155,200 + k is a prime times k, for k = 1..14
          > See http://www.primepuzzles.net/puzzles/puzz_181.htm
          > 5,516,280 + k is the smallest solution for k = 1..7
          >
          > > 34,415,168 is divisible by 2
          > > 34,415,169 is divisible by 3
          > > 34,415,170 is divisible by 5
          > > 34,415,171 is divisible by 7
          > > 34,415,172 is divisible by 11
          > > 34,415,173 is divisible by 13
          > > 34,415,174 is divisible by 17
          > > 34,415,175 is divisible by 19
          > > 34,415,176 is divisible by 23.
          >
          > This is what CRT (Chinese Remainder Theorem) is for. A bigint
          program can easily
          > reach primes in the millions but I will spare you solutions with
          millions of
          > digits.
          >
          > > 12,252,242 is divisible by 2
          > ...
          > > 12,252,258 is divisible by 18.
          >
          > Because lcm(2,...,18) = 2^4 * 3^2 * 5 * 7 * 11 * 13 * 17 =
          12,252,240
          > (lcm = least common multiple)
          >
          >
          > My contribution to fun with successive numbers is the smallest case
          of 10
          > numbers where the nth has n prime factors:
          >
          > 3931520917431241 = 3931520917431241
          > 3931520917431242 = 2 * 1965760458715621
          > 3931520917431243 = 3 * 221477 * 5917124453
          > 3931520917431244 = 2 * 2 * 23 * 42733923015557
          > 3931520917431245 = 5 * 17 * 199 * 5557 * 41826179
          > 3931520917431246 = 2 * 3 * 29 * 83 * 261799 * 1039837
          > 3931520917431247 = 7 * 7 * 7 * 19 * 43 * 557 * 25187741
          > 3931520917431248 = 2 * 2 * 2 * 2 * 31 * 59 * 167 * 804471071
          > 3931520917431249 = 3 * 3 * 3 * 3 * 3 * 41 * 53 * 1721 * 4326271
          > 3931520917431250 = 2 * 5 * 5 * 5 * 5 * 5 * 11 * 13 * 13 * 338377271
          >
          > The nth number is divisible by n above, but not in the 2nd smallest
          case
          > starting at 5,818,684,827,160,441.
          >
          > See http://www.research.att.com/projects/OEIS?Anum=A072875
          >
          > --
          > Jens Kruse Andersen
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