The Nth Gryphon Number
Jelly, 9 bytes

bṖ’ḅi-µ#Ṫ


A full program which reads a (1-indexed) integer from STDIN and prints the result.

Try it online!

How?

A Gryphon number is a number which is expressible in a base less than itself such that all the digits are ones except the least significant, which is a zero. For example:

\$30=1\times 2^4+1\times 2^3+1\times2^2+1\times2^1+0\times2^0 \rightarrow 30_2 = 11110\$

\$84=1\times 4^3+1\times 4^2+1\times 4^1+0\times 4^0 \rightarrow 84_4 = 1110\$

This program takes n, then starts at v=0 and tests for this property and increments v until it finds n such numbers, then outputs the final one.

To test a base b number it subtracts one from every digit, converts from base v, and then checks if the result is \$-1\$. (Note that b is less than v)

\$30_2 \rightarrow 0\times 30^4+0\times 30^3+0\times30^2+0\times30^1+(-1)\times30^0 = -1\$

\$84_4 \rightarrow 0\times 84^3+0\times 84^2+0\times 84^1+(-1)\times 84^0 = -1\$

bṖ’ḅi-µ#Ṫ - Main Link: no arguments
       #  - set v=0 then count up collecting n=STDIN matches of:
      µ   -  the monadic link -- i.e. f(v):  e.g. v=6
 Ṗ        -    pop (implicit range of v)            [1,2,3,4,5]
b         -    to base (vectorises)                 [[1,1,1,1,1,1],[1,1,0],[2,0],[1,2],[1,1]]
  ’       -    decrement (vectorises)               [[0,0,0,0,0,0],[0,0,-1],[1,-1],[0,1],[0,0]]
   ḅ      -    from base (v) (vectorises)           [0,-1,5,1,0]
     -    -    literal -1                           -1
    i     -    first index of (zero if not found)   2
          - }  e.g. n=11 -&gt; [6,12,14,20,30,39,42,56,62,72,84]
        Ṫ - tail         -&gt; 84
          - implicit print