Jonathan. Frech’s WebBlog

A278328 (#162)

Jonathan Frech,

The On-Line Encyclopedia of In­te­ger Sequences (also known by its acronym, OEIS) is a database hosting hundreds of thousands of — as the name implies — in­te­ger sequences. Yet, de­spite the massive num­ber of entries, I contributed a new in­te­ger sequence, A278328.

A278328 describes numbers whose absolute difference to their dec­i­mal re­verse are square. An example would be 12 or 21 (both are the dec­i­mal re­verse to each oth­er), since $\left|12-21\right|=9$$\left|12-21\right|=9$$\left|12-21\right|=9$ and $9=3^2$$9=3^2$$9=3^2$.

Not a whole lot is known about the sequence⁠¹, partly due to its definition on­ly resulting in the sequence when using the dec­i­mal system, though it is known that there are infinitely many numbers with said prop­er­ty. Since there are infinitely many palindromes (numbers whose re­verse is the num­ber itself), $\left|n-n\right|=0$$\left|n-n\right|=0$$\left|n-n\right|=0$ and $0=0^2$$0=0^2$$0=0^2$.

Due to there — to my knowledge — not being a direct formula for those numbers, I wrote a Python script to gen­er­ate them. On the sequence’s page, I posted a pro­gram which endlessly spews them out, though I later wrote a Python two-liner, which on­ly calculates those members of the sequence in the range from 0 to 98 (shown below entered in a Python shell).

>>> import math
>>> filter(lambda n:math.sqrt(abs(n-int(str(n)[::-1])))%1 == 0, range(99))
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 21, 22, 23, 26, 32, 33, 34, 37, 40, 43, 44, 45, 48, 51, 54, 55, 56, 59, 62, 65, 66, 67, 73, 76, 77, 78, 84, 87, 88, 89, 90, 95, 98]
[1][2020-07-29] “Don’t know”, you know?