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

[Alex M]

Speaking very roughly, if these are something like random distribution --
one would expect to find arithmetic sequences of 3 terms fairly easily, but
4 will be tricky, and for 5 or more would only occur finitely often (where
that finite number might be 0!). For the three-term sequences, if we write
them as (a, a+b, a+2b), then roughly the order of each term must be the
same -- they all have the same number of digits. If it's k digits, then we
have -- up to constant factor -- about 10^k possibilities for a and 10^k
possibilities for b. The resultant concatenation is of order 10^3k, which
is chosen random has a roughly 1 / 10^(3k/2) chance of being a perfect
square. Thus for a k we have -- again, up to a constant factor -- about
10^2k degrees of freedom, each with a 10^(-3k/2) chance of succeeding, for
a total of 10^(k/2) expected instances for each k. That is, the more
digits, the more solutions there are (finding them still becomes harder as
we go up, though.) A similar heuristic tells us that for 4 terms we'd find
a constant number for each k, for 5+ we'd find 10^(-nk/2) which when summed
over k converges.

Of course this relies on the notion that it will actually behave randomly,
but I get the feeling that "base" sequences usually do, accurately (but are
extremely difficult to prove so).

-Alex Meiburg

~6 out of 5 statisticians say that the
number of statistics that either make
no sense or use ridiculous timescales
at all has dropped over 164% in the
last 5.62474396842 years.
Looks interesting enough to be added to the OEIS - so please go ahead!

On Wed, Jul 2, 2014 at 11:21 AM, Christian Perfect
<christianperfect at gmail.com> wrote:
> 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?
>
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>
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