[seqfan] Re: About a problem of Bernardo Recamán Santos

[William Rex Marshall]

On 27/11/2010 1:01 a.m., Claudio Meller wrote:
> Hi seqfans,
> I found this problem in http://www.mathpuzzle.com/
> A problem of squaresBernardo Recamán Santos: Find a four-digit square number
> which has at least one digit in common with every other four-digit square
> number. [A nice little problem.]So I think about this sequence :
>
> Squares of k digits which has at least one digit in common with every other
> k-digit square number.
>
> 6241, 20164, 92416, 124609,  128164,  132496,  162409,  165649,  186624,

[cut]

In base 10, the smallest square which shares at least one digit with 
*every* other square (greater than zero), not just those with the same 
number of digits, is 92416. If we also include 0 as a square number, 
then 124609 is the smallest square.

Why is this? Because the last nonzero digit of any positive square is 
either 1, 4, 5, 6 or 9, and if 5 is the last nonzero digit, then 2 is 
the second-last nonzero digit.

The sequence of squares which share at least one digit with every other 
positive square begins:

92416, 124609, 132496, 162409, 165649, 195364, 196249, 214369, 264196, 
346921, 351649, 395641, 401956, 436921, 495616, 541696, 543169, 651249, 
669124, 1046529, 1192464, 1240996, 1247689, 1290496, 1354896, 1406596, 
1456849, 1459264, ...