Problems & Puzzles:
Puzzles
Puzzle 283.
Harshad Left Truncatable numbers
Jason Earls calculated very
recently (August 28, 2004) the sequence
A097569
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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