Thursday, March 15, 2012

A Puzzle

Still here at the van der Poorten memorial conference in Newcastle, Australia. The following problem occurred to me:

What is special about the integer 2007986541?

The labels for the post may give some hints.
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10 comments:

  1. Bayesian Bouffant, FCD9:46 AM, March 15, 2012

    It lacks a '3'

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  2. George2:24 PM, March 15, 2012

    The sum of the digits is the answer to the epoch question - What is the meaning of life the universe and everything -

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  3. George9:58 PM, March 15, 2012

    Okay a bit more, its divisible by the sum of its digits as well.

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  4. Miranda1:21 AM, March 16, 2012

    It's a Harshad number. These numbers give a person joy.

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  5. George5:21 PM, March 16, 2012

    It is NOT divisible by the sum of its digits! The sum of its digits is 42 and its prime factorization is 3 x 11 x 60848077. I don't see what is special about it yet.

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  6. Jeffrey Shallit7:03 PM, March 16, 2012

    Hint: try the sum of digits in some prime bases.

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  7. Jeffrey Shallit7:36 PM, March 16, 2012

    Miranda's a real googlin' fool! Knows no more or less than google can provide.

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  8. g8:15 PM, March 16, 2012

    Sum of digits is 21 in bases 2,3,5,7,11,13. (So it's Harshad in bases 3 and 11, for what that's worth.)

    Naive searching quickly finds that 1386, 1387, 485353 have digit-sums the same in bases 2,3,5,7,11. Is 2007986541 the smallest such number when base 13 is included too? (My slow-and-dirty searching program says there are none below 20,000,000. I was too impatient to let it run >100x longer.)

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  9. Jeffrey Shallit11:57 AM, March 17, 2012

    Right, it's the smallest integer > 1 whose sum of digits in bases 2,3,5,7,11, and 13 is the same.

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  10. Miranda9:27 PM, March 17, 2012

    "Knows no ... less than google can provide."

    Thanks for the compliment!!

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