The Nth Gryphon Number I came up with a series of numbers the other day and decided to check what the OEIS number for it was.  Much to my surprise, the sequence did not appear to be in the OEIS database, so I decided to name the sequence after myself (note that someone else who's a lot smarter than me has probably already come up with this, and if someone finds the actual name of this sequence, please comment and I'll change the question title).  As I couldn't find the sequence anywhere, I decided to name it after myself, hence "Gryphon Numbers".  EDIT: Thanks to @Surb for bringing to my attention the fact that this sequence is equal to OEIS sequence A053696 - 1.  

A Gryphon number is a number of the form \$a+a^2+...+a^x\$, where both \$a\$ and \$x\$ are integers greater than or equal to two, and the Gryphon sequence is the set of all Gryphon numbers in ascending order.  If there are multiple ways of forming a Gryphon number (the first example is \$30\$, which is both \$2+2^2+2^3+2^4\$ and \$5+5^2\$) the number is only counted once in the sequence.  The first few Gryphon numbers are: \$6, 12, 14, 20, 30, 39, 42, 56, 62, 72\$.

Your Task:

Write a program or function that receives an integer \$n\$ as input and outputs the \$n\$th Gryphon number.  

Input:

An integer between 0 and 10000 (inclusive).  You may treat the sequence as either 0-indexed or 1-indexed, whichever you prefer.  Please state which indexing system you use in your answer to avoid confusion.  

Output:

The Gryphon number corresponding to the input.     

Test Cases:

Please note that this assumes the sequence is 0-indexed.  If your program assumes a 1-indexed sequence, don't forget to increment all the input numbers.  

Input:    Output:
0   ---&gt;  6
3   ---&gt;  20
4   ---&gt;  30
10  ---&gt;  84
99  ---&gt;  4692
9999 --&gt;  87525380


Scoring:

This is code-golf, so the lowest score in bytes wins.
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