[seqfan] Numbers x such that the base 10 representation of x^2 forms an arithmetic sequence when split into equal-sized chunks

[Christian Perfect]

The twitter feed @onthisdayinmath tweeted the fact that 183^2 = 183184.
This leads immediately to A030467. I came up with the following questions:

- are there any x such that x^3 = (a)(a+1)(a+2)?
- are there any x such that x^3 = (a)(a-1)(a-2)?
- are there any x such that x^3 = (a)(a)(a)?

The answer to all of those, as far as I can see, is "no, for for x <
10000000". A disclaimer: I'm a middling mathematician and I haven't come up
with any reasons why these sequences might not exist.

So, I decided to widen my search: are there any (x,n), with n>2, such that
the base 10 representation of x^n forms an arithmetic sequence when split
into three or more equal-sized chunks? The answer to that also appears to
be "no, for x < a fairly large number". I wonder if I'm just asking for
something so specific that I need to look at orders of magnitude more
candidates.

Anyway, in defeat, I decided to see if I could get numbers whose squares
form arithmetic sequences when you split them into three or more
equal-sized chunks. I got the following:

11142,11553,14088,16713,18801,22284,23097,23718,26787,28818,323589,327939,328992,416103,438357,459069,
...

For example, 11142^2 = 124144164, and 124, 144, 164 is an arithmetic
sequence.

This doesn't seem to be in the OEIS, but my route to it was so convoluted
that I'm not sure whether it's worth adding. By the way, these all split
into three chunks - I haven't found a number yet which gives an arithmetic
sequence of 4 chunks.

So, should I add the above sequence?