## 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.

Bayesian Bouffant, FCD said...

It lacks a '3'

George said...

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

George said...

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

Miranda said...

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

George said...

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.

Jeffrey Shallit said...

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

Jeffrey Shallit said...

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

g said...

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.)

Jeffrey Shallit said...

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

Miranda said...

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

Thanks for the compliment!!