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rare numbers
Shyam Sunder Gupta calls a number  $n$  rare if  $n+r$  and  $n-r$  are both squares, where  $r\neq n$  is the reversal of  $n$.

For example, 621770 is rare because  $621770+77126 = 836^2$  and  $621770-77126 = 738^2$.

According to Gupta there are only 84 such numbers up to  $10^{20}$.

The first rare numbers are 65, 621770, 281089082, 2022652202, 2042832002, 868591084757, 872546974178, 872568754178 more terms

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here
remainders of rare numbers

A graph displaying how many rare numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.
divisibility of rare numbers
Useful links Shyam Sunder Gupta, Rare Numbers
OEIS, Sequence A035519
Rare numbers can also be... (you may click on names or numbers)

aban 65 alternating 65 amenable 65 arithmetic 65 congruent 65 Cullen 65 Cunningham 65 Curzon 65 cyclic 65 deficient 65 621770 Duffinian 65 esthetic 65 evil 65 281089082 happy 621770 inconsummate 65 katadrome 65 magic 65 magnanimous 65 nialpdrome 65 oban 65 octagonal 65 odious 621770 pernicious 65 621770 Proth 65 semiprime 65 sliding 65 uban 65 wasteful 65 621770