9 billion names of God the integer

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This task is a variation of the short story by Arthur C. Clarke.

(Solvers should be aware of the consequences of completing this task.)

In detail, to specify what is meant by a "name":

  • The integer 1 has 1 name "1".
  • The integer 2 has 2 names "1+1" and "2".
  • The integer 3 has 3 names "1+1+1", "2+1", and "3".
  • The integer 4 has 5 names "1+1+1+1", "2+1+1", "2+2", "3+1", "4".
  • The integer 5 has 7 names "1+1+1+1+1", "2+1+1+1", "2+2+1", "3+1+1", "3+2", "4+1", "5".

This can be visualized in the following form:

      1
    1   1
  1   1   1
1   2   1   1

1 2 2 1 1 1 3 3 2 1 1

Where row $n$ corresponds to integer $n$, and each column $C$ in row $m$ from left to right corresponds to the number of names beginning with $C$.

Optionally note that the sum of the $n$-th row $P(n)$ is the integer partition function.

Test

{{test}}

Console output