[seqfan] Re: n and n^2 use digits 0, 1, 4, 6 only: A136859 - a question

[M. F. Hasler]

On Fri, Jan 29, 2016 at 7:35 AM, Neil Sloane <njasloane at gmail.com> wrote:

> A136859 contains the numbers n such that n and n^2 use only the digits
> 0,1,4,6.
>
> It appears that n must be 0, 10^k, or 4*10^k. Does anyone know if this is a
> theorem?
>
The entry had a formula which was based on there being no other type of
> term, but this was only a conjecture.  I deleted the program and the b-file
> that were based on the conjecture.
>

I think nonetheless that such formulas and programs might be left,
provided they are clearly marked as (based on a) conjecture.

Thanks to Maximilian for raising suspicions about the formula.
>

I was just wondering because I know of some other similar cases where
"sporadic" solutions exist.

It would be nice if someone could check that the terms shown are complete,
> that there are no missing numbers. The terms shown are the original ones
> found by Jonathan Wellons, so they are probably correct, but it would be
> good to have a check.
>

Up to the listed terms this is correct ;
I did not have the time to really think about it but I quickly wrote a
program that computes the 10-adic expansion for candidates for possible
nontrivial solutions (they necessarily have to end with "6")
one early interesting candidate being
....000000140440444461046
with square
....8440416146101411414116 ;
It further extends to ...61664000000140440444461046 (26 digits)
with a square having its first "forbidden digit" at 10^29,
or ...4141414000000140440444461046 whose square is OK up to 10^32
and so on.

Of course this is neither a proof nor a counter-example...

Maximilian
PS: I think that "Numbers a(n) such that ..." (which can be found
elsewhere) should be avoided, it is in some sense incorrect, because it
does not exclude [and thus in some sense even suggest] that there are other
numbers, not listed as a(n), that also have the given property.