[seqfan] Re: Squares with even-positioned digits matching the original number

[W. Edwin Clark]

The analogous sequence for base 2 has lots of small values easily computed:

1,2,4,5,8,10,16,17,20,21,32,34,40,42,64,65,68,69,80,81,84,128,130,136,138,160,162,168,256,257,260,261,272,273,276,277,320,321,324,336,337,512,514,520,522,544,546,552,554,640,642,648,672,674,1024,1025,1028,1029,1040,1041,1044,1045,1088,1089,1092,1093,1104,1108,1280,

For example, let r(n) be the base 2 representation of n then
r(20) = [1, 0, 1, 0, 0]
r(400 )= [1, 1, 0, 0, 1, 0, 0, 0, 0]

On Fri, Dec 1, 2023 at 2:54 PM David Radcliffe <dradcliffe at gmail.com> wrote:

> Hi all,
>
> I recently came across sequence A326418, which is described as "Nonnegative
> numbers k such that, in decimal representation, the subsequence of digits
> of k^2 occupying an odd position is equal to the digits of k." I was
> wondering about the analogous problem for even positions, but that sequence
> is not in the OEIS.
>
> I performed a search up to 10^16, and the only example I found was k =
> 9678692507732, excluding multiples of 10.
>
> This is a term because k^2 = 93677088659227550879783824, and if we remove
> every other digit from k^2, starting with the units digit, we get back to
> 9678692507732.
>
> Are there other solutions? I am intrigued by this sequence because the
> first term is so large, but I don't know enough terms to propose it for
> inclusion in the OEIS.
>
> - David
>
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