What is the (apparently) next number in the sequence?

$$42, 62, 67, 14, 27, 45, 4, 14\dots$$

This is the number sequence which does not appear in OEIS and actually has little to do with maths. To find the next number, please look at the title and the tags!

Hint 1:

TREN (remember the code which is most frequently used here). Note that the both 14s in the sequence are in fact very different!

Hint 2:

The sequence resides in a (current) European country which didn't exist at the time when the sequence started.

Hint 3:

In the other sequences of this kind, the numbers are mostly single-digit, and almost never higher than 23.

Hint 4 (maybe decisive):

For every sequence $$\{r_k\}$$ of this kind, we usually can define a function $$f:\mathbb{N}\to\mathbb{N}$$, such as for all $$k$$ exist such indices $$i_1 where $$f(r_{i_1})=f(r_{i_2})=\dots=f(r_{i_n})$$ and $$r_{i_j}=j$$. (i.e. $$r_{i_1}=1,r_{i_2}=2$$ etc. up to $$r_{i_n}=r_k=n$$).
But there are exceptions. One of the most famous sequences of this kind fails on $$r_{187}=21$$, where there is no $$i_{20}$$ to define.
P.S. Despite strong mathematical language, the rule is actually quite simple.

• I think I know what are these numbers, but I don't know WHY are them... rot13(Vg'f fbzrguvat eryngrq gb jnef)? Feb 27 '20 at 13:29
• @MatíasRodríguez Unfortunately no, but you're probably on the right track. I'm adding another hint. Feb 27 '20 at 16:44
• The hint 1 is in code? Feb 27 '20 at 17:37
• Yes, it's encoded. Feb 27 '20 at 18:56
• 3rd hint added. Feb 29 '20 at 13:46

29

The sequence is given by:

The regnal numbers of the princes of the House of Reuss: Heinrich XLII, Heinrich LXII, Heinrich LXVII, Heinrich XIV, Heinrich XXVII, Heinrich XLV, Heinrich IV, and Heinrich XIV. The heir apparent of the principality is Heinrich XXIX, hence the answer is apparently 29.

Hint 1 refers to:

The rot13 of GERA, which is a German city under the rule of the House of Reuss.

Hint 2 refers to:

The fact that Germany was not a unified country when the sequence began with Prince Heinrich XLII in 1806 (the German Confederation was formed in 1815).

Hint 3 refers to:

The fact that most monarchies contain regnal numbers which are based on the sequence of the ruling heads of house, which means the numbers do not get very large. The regnal numbers in the House of Reuss, however, are assigned to every male family member upon birth resetting in each century, meaning the regnal numbers for the ruling princes can become quite large and sporadically spaced.

Hint 4 refers to:

The fact that in most monarchies, you can find a sequential regnal numbering for any given name (i.e. there is an Edward I, Edward II, etc.). This is not the case in the House of Reuss for the reasons stated above. The famous sequence referred to in the hint is the numbering of Pope John, which skipped Pope John XX.

• Great! You finally got it. Mar 5 '20 at 17:59
• P.S. Bonus question: what's the "famous" sequence from hint 4? Mar 5 '20 at 17:59
• @trolley813 Updated with answer to your bonus question! Mar 5 '20 at 18:06
• Excellent answer! Congratulations Mar 5 '20 at 18:10

The answer is based acording my investigations, but maybe it would fail:

Let me explain something:

The numbers are the king's names on Europe. That's why the author speaks about list of this kind, these lists are the list of the individual name of kings.

This helped me

The exception that he talks is maybe for the king Luis XVII of France, who never was really a king, but his successor choose the name of Luis XVIII when he should be XVIII.