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".

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   --->  6

3   --->  20

4   --->  30

10  --->  84

Scoring:

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

MATL, 16 13 bytes

:Qtt!^Ys+uSG)

1-based.

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Explanation

Consider input n = 3 as an example.

:    % Implicit input: n. Range

     % STACK: [1 2 3]

Q    % Add 1, element-wise

     % STACK: [2 3 4]

tt   % Duplicate twice, transpose

     % STACK: [2 3 4], [2 3 4], [2;

                                 3;

                                 4]

^    % Power, element-wise with broadcast

     % STACK: [2 3 4], [ 4   9  16;

                         8  27  64;

                        16  81 256]

Ys   % Cumulative sum of each column

     % STACK: [2 3 4], [ 4    9  16;

                         12  36  80;

                         28 117 336]

+    % Add, element-wise with broadcast

     % STACK: [ 6  12  20;

               14  39  84

               30 120 340]

u    % Unique elements. Gives a column vector

     % STACK: [  6;

                14;

                30;

                12;

               ···

               340]

S    % Sort

     % STACK: [  6;

                12

                14;

                20;

               ···

               340]

G)   % Push input again, index. This gets the n-th element. Implicit display

     % STACK: 14

Wolfram Language (Mathematica), 59 bytes

Union[Join@@Table[Sum[n^k,{k,j}],{j,2,30},{n,2,7!2}]][[#]]&

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1-indexed

Jelly, 9 bytes

bṖ’ḅi-µ#Ṫ

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

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