Index in PRNG output sequence
Pick your favorite $b$-bit block cipher $E_k$, say AES-256.  Let $k$ be the seed and let the $i^\text{th}$ output be the $b$-bit string $x_i = E_k(i)$.  Then $i = E_k^{-1}(x_i)$.  Given $k$ and $x_i$, this is widely considered to be ‘easy’, for most block ciphers including AES-256.  Note that every possible $b$-bit string has an index.

$\newcommand{\Z}{\mathbb{Z}}$Pick your favorite 2048-bit safe prime $p$, say $2^{2048} - 1942289$, and your favorite nonzero generator $a$ of $(\Z/p\Z)^\times$, say $5$.  Let $x_0 \in (\Z/p\Z)^\times$ be the seed and let the $i^\text{th}$ output be $$x_i = a x_{i - 1} = a^i x_0.$$  Then $a^i = x_i x_0^{-1}$, so that $i = \log_a x_i x_0^{-1}$.  This is widely considered to be ‘hard’, for sufficiently large safe primes including $2^{2048} - 1942289$.  Note that every possible element of $(\Z/p\Z)^\times$ has an index.

Defining ‘easy’ and ‘hard’ is left as an easy exercise for the reader.  Deciding whether to encode $i$ as little-endian or big-endian for $E_k$ is left as a hard exercise for the reader.

This does not address any security of these PRNGs.  One of these is fit for use in cryptography; the other is emphatically not.  Since you didn't mention any security requirements in your question, I didn't mention any security properties in my answer.  Figuring out which one is which is also left as an exercise for the reader.  (I hope you're taking notes, because there'll be a quiz on Thursday next week about all these exercises!)