Problems & Puzzles: Puzzles

Puzzle 283.  Harshad Left Truncatable numbers

Jason Earls calculated very recently (August 28, 2004) the sequence about the zero free, Harshad right tuncatable numbers. The largest of these numbers is 24786.

24786 is divided by 27
2478 is divided by 21
247 is divided by 13
24 is divided by 6 and
2 is divided by 2

Very naturally I wanted to know what was the largest zero free, Harshad left truncatable number, and I got a surprise... but I will not spoil your own fun.

Questions:

1. Calculate the total quantity of zero free Harshad left truncatable numbers

2. Calculate the ten (10) largest zero free Harshad left truncatable numbers

3. Can you explain why the largest of these is so big compared to 24786? Contributions came from Luke Pebody, Jon Wharf and J. van Delden:

Luke wrote for the question 3:

It seems part of the reason is because 10 (the base in question) is not prime.

If we let l(b) be the largest Harhsad left truncatable number and r(b) be the largest Harshad right truncatable number, it seems that l(b) is much larger if b is a prime than if it is not, whereas r(b) seems to profit from having many factors. I haven't even a heuristic argument for why this is so, but if the base b is less than 20 (apart from some unusual behavior at b=5):

If b is prime, r(b)<6l(b)
If b is not prime, r(b)>6l(b).

The extreme bases up to 20 are:

Base 18, r(b)=4863F2H2F22851GA48C88888925GDD6429B6 and l(b)=6B8A Base 17, r(b)=AF3B9G6933BCD393C25311 and l(b)=311553663339F9664C5FAAEE.

Can't give you any reasons for this, though.

Jon Wharf and Jan van Delden wrote the same results for Questions 1 & 2:

There are 3594 Harshad left-truncatable numbers (not including the 9 single digits).

The ten largest are:
8784581777424862253312
8798172666441924314112
9699636934134242462112
84399636934134242462112
684399636934134242462112
884399636934134242462112
4884399636934134242462112
44884399636934134242462112
64884399636934134242462112
364884399636934134242462112

Jon added for the Question 3

I think the reason there are so many left-truncatable numbers compared to the right-truncatable variety is clear when you try to build up such numbers by adding digits to the left. The divisibility by powers of two is preserved as digits are added to the left-hand end. This does not happen when building up the right-truncatable Harshad numbers (of which there are only 26).

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