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sliding numbers
A number  $n$  is called sliding if there exists two numbers  $x+y=n$  such that
\[
    \frac{1}{x}+\frac{1}{y} = \frac{n}{10^k}
\]
for some  $k\ge0$. Note that  $n/10^k$  is simply  $n$  "shifted" by  $k$  decimal places.

For example, 7 is a sliding number because  $7=2+5$  and  $1/2+1/5=0.7$. Similarly, 646625 is a sliding number because  $256000+390625=646625$  and

\[
\frac{1}{256000}+\frac{1}{390625}=0.00000646625\,.
\]

In practice, a number  $n$  is sliding if it can be written as  $x+10^k/x$  for some  $k\ge 0$, where  $x$  is a divisor of  $10^k$. The sliding numbers which are not divisible by 10 are called primitive sliding numbers.

Up to  $10^{18}$  there are only 3861 sliding numbers.

The first sliding numbers are 2, 7, 11, 20, 25, 29, 52, 65, 70, 101, 110, 133, 200, 205, 250, 254, 290, 425, 502, 520, 641, 650 more terms

Sliding numbers can also be... (you may click on names or numbers and on + to get more values)

a-pointer 11 101 ABA 200 2500 20000 + 2000000000000 aban 11 20 25 + 925000000000 abundant 20 70 200 + 42500000 Achilles 200 2000 20000 + 20000000000000 admirable 20 70 650 alt.fact. 101 alternating 25 29 52 + 490329250 amenable 20 25 29 + 986802500 amicable 5020 apocalyptic 650 700 925 + 29000 arithmetic 11 20 29 + 9868025 automorphic 25 Bell 52 binomial 20 70 1001 1330 brilliant 25 c.decagonal 11 101 c.octagonal 25 c.square 25 925 3445 Chen 11 29 101 641 congruent 20 29 52 + 9250000 constructible 20 Cullen 25 65 Cunningham 65 101 290 + 100000000000001 Curzon 29 65 254 + 97666490 cyclic 11 29 65 + 9766649 d-powerful 2225 22025 62660 + 3231250 de Polignac 10001 48830173 decagonal 52 deceptive 10001 100001 100000001 10000000001 deficient 11 25 29 + 9868025 dig.balanced 11 52 290 + 195824500 Duffinian 25 65 133 + 9868025 eban 52 2000 2050 + 52000000000000 economical 11 25 29 + 20000000 emirpimes 205 502 16265 + 3906506 enlightened 250 2500 25000 + 250000000000 equidigital 11 25 29 + 16265000 eRAP 20 esthetic 65 101 1010 Eulerian 11 502 evil 20 29 65 + 1000000001 Friedman 25 2500 2504 + 393185 frugal 20000 25000 160625 + 925000000 gapful 110 200 700 + 81250000000 Gilda 29 110 good prime 11 29 101 641 happy 70 133 700 + 9775865 Harshad 20 70 110 + 9250000000 hoax 250 650 2500 + 78500000 Hogben 133 iban 11 20 70 + 700000 idoneal 25 70 133 520 inconsummate 65 650 785 + 785000 interprime 205 254 650 + 97657274 Jacobsthal 11 junction 101 925 1001 + 97758650 katadrome 20 52 65 + 6410 Lehmer 133 Leyland 20000000000 Lucas 11 29 lucky 25 133 205 + 7073125 magic 65 magnanimous 11 20 25 + 2225 metadrome 25 29 1258 15689 modest 29 133 6266 Moran 133 15689 90925 1563140 nialpdrome 11 20 52 + 700000000000000 nude 11 O'Halloran 20 oban 11 20 25 + 925 octagonal 65 133 odious 11 25 52 + 976664900 palindromic 11 101 1001 + 100000000000001 palprime 11 101 pancake 11 29 254 panconsummate 11 20 pandigital 11 16265 200432500 partition 11 101 pentagonal 70 425 925 1001 pernicious 11 20 25 + 9868025 Perrin 29 plaindrome 11 25 29 + 15689 power 25 2500 250000 + 25000000000000 powerful 25 200 2000 + 250000000000000 practical 20 200 520 + 9250000 prim.abundant 20 70 650 prime 11 29 101 641 pronic 20 110 650 + 100000010000000 Proth 25 65 641 pseudoperfect 20 200 520 + 925000 rare 65 repdigit 11 repunit 133 Ruth-Aaron 25 1330 self 20 110 3157 + 250000004 semiprime 25 65 133 + 10000001 Smith 62660 sphenic 70 110 290 + 48830173 square 25 2500 250000 + 25000000000000 strobogrammatic 11 101 1001 + 100000000000001 strong prime 11 29 101 641 super Niven 20 70 110 + 20000050000 super-d 15689 78253 156890 + 7825300 tau 2000 2504 25040 + 801250000 tetrahedral 20 1330 tetranacci 29 trimorphic 25 truncatable prime 29 twin 11 29 101 641 uban 11 20 25 + 70000000000000 Ulam 11 502 5020 + 6415625 undulating 101 1010 unprimeable 200 1330 6410 + 9775865 untouchable 52 290 520 + 785000 wasteful 20 52 65 + 9868025 weird 70 Zuckerman 11 Zumkeller 20 70 520 + 92500 zygodrome 11 1100 11000 + 110000000000000