That you can find other special properties on various numbers is true in all bases, but 1/7 in base 10 is pretty special.
So the property it has is:
1/n = n (2/b^2 + 4/b^4 + 8/b^6 + ...)
which by geometric series sums to
1/n = n / (b²/2 — 1)
n² = b²/2 – 1
So this works precisely because 7² = 49 = 50 – 1 = 100/2 — 1.
Calculating some of these out these appear to be the Newman-Shanks-Williams numbers [1], the next one is 41 in base 3364, where
1/41 = {0}.{82}{164}{328}{656}{1312}{2625}...
notice the 5 finally coming from some overflow.
But, supposing that we just like the idea of starting with some digit d and then the next digit being k times that and the next digit being k times that, we get a more general set of numbers,
d/b + dk/b² + dk²/b³ + ...
= d/(b - k)
Given that, this becomes much more boring. So for example for doubling in base-100 we think about 1/98 (b=100, k=2) and we find
1/98 = 0.01020408163265...
and factors of that 98 also may have similar patterns, so 7 has this strength because it is a factor of 98.
So for example we want to think about 1/7 in base-12, this suggests that maybe we should look for things that quintuple base 12, but that rapidly overflows base 12. So we do the same trick as 1/7 where we take pairs of digits, and maybe things quadruple base-144 (since 144 - 4 is 140 which is divisible by 7), and so we find that
1/7 = 0.{20}{82}{41} repeating
and if you squint closely you can see starting with 20, quadrupling to 80, quadrupling to 320 but then getting a bit unwieldy. Of course even on single digits 12 - 2 = 10 which has 5 as a factor so you can expect to see a pattern in base-12 on
1/5 = 0.{2}{4}{9}{7} [repeating]
which you can see a sort of "2, 4, 8, 16," pattern happening.
The other base that I really like is nonnary, if we met aliens we might find that they count in balanced nonnary with digits -4, -3, -2, -1, 0, 1, 2, 3, 4, (so like 7 is actually {1, -2}, 7 = 9 - 2), but it's harder to search for patterns in that because you really feel the cap of having only half the base to count up to before you carry.
