Gilbreath's Conjecture Suppose we start with the infinite list of prime numbers:

[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, ...


Then, we take the absolute differences between each pair of numbers, repeatedly:

[1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, ...
[1, 0, 2, 2, 2, 2, 2, 2, 4, 4, 2, 2, 2, 2, 0, 4, 4, 2, ...
[1, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 4, 0, 2, ...
[1, 2, 0, 0, 0, 0, 2, 2, 2, 2, 0, 0, 2, 2, 4, 2, ...


Notice that the leading number is 1 every time. Gilbreath's Conjecture is the prediction that this continues to be the case forever.

The only way the leading number would stop being a 1 is if the next number after it was neither a 0 nor a 2. The only way the second number wouldn't be a 0 or a 2 is if the number after that was neither a 0 nor a 2. And so on.

The index of the earliest number, other than the leading 1, which is neither a 0 nor a 2, can never go down by more than 1 between a consecutive pair of sequences. This fact has been used to put a very strong lower bound on when, if ever, a sequence might not have a 1 as the first element.

In this challenge, you will be given the index of a sequence, and you must output the index of the first number in that sequence which is not the leading 1, and is not a 0 or a 2.

For instance, in the 4th absolute difference sequence above:

[1, 2, 0, 0, 0, 0, 2, 2, 2, 2, 0, 0, 2, 2, 4, 2, ...


The first entry that's neither a zero or a two, other than the first entry, is the 15th position, 14 zero indexed. So if the input was 4, you would output 14.

For inputs from 1 to 30, the outputs should be:

[3, 8, 14, 14, 25, 24, 23, 22, 25, 59, 98, 97, 98, 97, 174, 176, 176, 176, 176, 291, 290, 289, 740, 874, 873, 872, 873, 872, 871, 870]

This is OEIS A000232.

This is assuming you have 1 indexed inputs and 0 indexed outputs. You may index your inputs and outputs starting at any constant integers, as long as you can accept the range of inputs corresponding to all sequences.

Requirements: Your solution must run in at most 1 minute on a input of up to 30. If it's close enough that it depends on the computer specs, it's allowed.

Shortest code wins.
Mathematica, 66 bytes

(For[z=1,Last@Nest[Abs@*Differences,Array[Prime,z+#],#]&lt;3,z++];z)&amp;


Pure function taking a positive integer as argument and returning a 1-indexed integer. Nest[Abs@*Differences,Array[Prime,z+#],#] computes the #th iterated absolute difference list of the list of the first z+# primes. For[z=1,Last@...&lt;3,z++] loops this computation until the last element of the resulting list is at least 3, and then z is output. (Note that the correctness of the algorithm assumes Gilbreath's conjecture!)