Crack the Lost Numbers

Question

The Lost Numbers are 4, 8, 15, 16, 23, 42. Your goal is to create an expression to output them all.

Your allowed characters are 0123456789, ^+-*/, (), % for modulo and n as the single variable allowed. Modified PEMDAS ()^*/%+- is the precedence and / is normal division (3/2 = 1.5). 0^0 = 1 although division by 0 is undefined.

Create the shortest expression (in characters) such that for n = 0..5 (or 1..6 for 1-indexing) the nth Lost Number will be the result of the expression. Shortest expression wins.

Answer 1

20 characters

Assuming that the precedence of the operators is similar to JS. The input is expected to be 1-indexed.

n*4+16%n*3+6*0^(6-n)

Try it online! (interpreted in JS ES7)

Breakdown

 n           |  1 |  2 |  3 |  4 |  5 |  6 | (x4)
 16 % n      |  0 |  0 |  1 |  0 |  1 |  4 | (x3)
 0 ^ (6 - n) |  0 |  0 |  0 |  0 |  0 |  1 | (x6)
-------------+----+----+----+----+----+----+------
             |  4 |  8 | 15 | 16 | 23 | 42 |

Answer 2

16 15 characters

6704196%(n*8+3)

One-indexed. A simple application of a generalisation of the Chinese Remainder Theorem, with the formula n*8+3 chosen to make it work (that is, where moduli have a common factor, the corresponding values' difference is also divisible by that factor), as well as to get a shorter number on the left.

Previously: 485559068%(n+41), zero-indexed

Answer 3

15 bytes

n*4+703%n^3%7*3

Try it online!

Based off @Arnauld's answer.

The idea was to look for a magic formula A%n^B%C that maps (1, 2, 3, 4, 5, 6) to (0, 0, 1, 0, 1, 6) to combine the last two terms in Arnauld's formula into one:

 n           |  1 |  2 |  3 |  4 |  5 |  6 | (x4)
 703%n^3%7   |  0 |  0 |  1 |  0 |  1 |  6 | (x3)
-------------+----+----+----+----+----+----+------
             |  4 |  8 | 15 | 16 | 23 | 42 |