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Re: [PrimeNumbers] Fun with (eeek!) composites

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  • 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 1 of 3 , Apr 4 4:13 PM
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      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 2 of 3 , Apr 4 6:30 PM
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        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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