There's an algorithm for every finite sequence. The algorithm for this particular sequence is:

Print 4

Print 8

Print 15

Print 16

Print 23

Print 42

Halt

Turing would approve.

I do take your point that there are sequences whose generating principle is social as opposed to mathematical. For example there's the famous sequence: 4, 14, 23, 34, 42, 50, 59, ... These are the northbound numbered stops of Manhattan's A train.

Absolutely. In his famous and witty pamphlet,

Mathematics Made Difficult, Carl Linderholm makes this exact same point.

Any finite sequence can be fitted to a

Lagrange interpolation polynomial. Since a polynomial is one of the simplest and most common mathematical functions, Linderholm points out that that

any number whatsoever is a correct response to any of these kinds of puzzles.

These questions don't test for mathematical ability; but rather for the ability to determine what the examiners think the right answer should be. They're testing for conformance and lack of creativity.

Even these "what number doesn't belong" are the same. All numbers belong and all numbers don't belong for a wide variety of interesting and uninteresting reasons.