[seqfan] Re: A070837

[Giovanni Resta]

On 07/23/2015 11:35 AM, Christian Perfect wrote:
> I've been computing a sequence which turned out to be A070837 - the
> smallest k such that |k-R(k)|=9n, or 0 if no such k exists.
> I found an error in a(21) - it was listed as 190, but |190-R(190)| is 9*11.
> I've corrected it to 1090.
> The comment by Sascha Kurz appears to be incorrect - should it say "at
> least 5 digits"? 9n=18, k =13 is the first of many counterexamples to the
> statement as I interpret it. The statement is true when changed to "at
> least 5 digits".
> We've compute values up to k=10^8, so my maximum n for which I'm sure of
> a(n) is 11108889, but before I submit a b-file I want to check that the
> zeroes in the published sequence are really zeroes. Can anyone help? The
> first zero is at n=12. A scatter plot of the values I've computed strongly
> suggests to me that every n has a non-zero k:
> http://checkmyworking.com/misc/a070837.png

Kurz comment is not clear to me.

I hope you will not submit a b-file with 11108889 elements...
Apart for sequences of exceptional significance, the usual upper limit
for a b-file is 10000 entries.

Checking the zeros is not difficult, the only things to do is estimate
the largest possible k that can give |k-R(k)|=9n.

It is easy to see that, if k has respectively 2, 3, 4, 5, 6, 7, 8, digits
the minimum nonzero value of |k-R(k)| is respectively equal to

2   9     (10 - 01)
3   99    (201 - 102)
4   90    (1101 - 1011)
5   990   (11001 - 10011)
6   900   (101001 - 100101)
7   9900  (1010001 - 1000101)
8   9000  (10010001 - 10001001)
9   99000 (100100001 - 100001001)
10  90000 (1000100001 - 1000010001)

a pattern clearly emerges.

This table helps understanding the upper bound for k when searching for 
a certain 9n.
For example, if we are interested in all
the n < 10000, i.e., 9*n <= 89991,
then we know that it is sufficient (and probably necessary) to
test all k with up to 8 digits, i.e. k < 10^8.
Indeed, k with 9 or 10 digits are not needed, since the minimum 
difference for 9 digits is 99000 > 89991, and for 10 digits is 90000  > 
89991. (For 11 or more digits we expect the minimum difference to be 
even larger).

so, to check the 76 values in Data representing differences up to 9*76 = 
684, checking k up to 9999 is enough.

Giovanni