TITLE: Rational approximation of pi
QUESTION [10 upvotes]: I found this problem intriguing: $355 / 113 = 3.14159292035398\ldots$ gives the approximation of $\pi$ in $7$ correct numbers, say $C(355/113)=7$, but it number of digits in numerator + number of digits in denominator is six, say $L(355/113)=6$. How many rationals $a/b$ there are such that $L(a/b)<C(a/b)$?

REPLY [2 votes]: Here is a few OEIS sequences which may help answer this question:
A002485 https://oeis.org/A002485
A002486 https://oeis.org/A002486
I hope this helps.