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Finding the root of a basic sequence

Puzzling Asked on May 20, 2021

The sequence starts with the following:

a1 = 1, a2 = 2, a3 = 5, a4 = 4, a5 = 6,
a6 = 10, a7 = 9, a8 = 8, a9 = 21, a10 = 12,

a11 = 13, a12 = 20, a13 = 33, a14 = 15, a15 = 42,
a16 = 16, a17 = 19, a18 = 63, a19 = 34, a20 = 24, …

Identify the rule of this sequence and the next five terms.


The same list of numbers, without indices:

1, 2, 5, 4, 6, 10, 9, 8, 21, 12, 13, 20, 33, 15, 42, 16, 19, 63, 34, 24

Hints are spoilered so that people can choose to solve it without seeing them.

Hint 1

Hint 2

Hint 3-1 (continuation of Hint 1)

Hint 3-2 (continuation of Hint 2)

2 Answers

It is very clear that -

Now,

Further,

Next,

Other sequences

From here I got confused and couldn't get it

Answered by AnilGoyal on May 20, 2021

Before I start, consider function $f$ that satisfies $A_x = f(x)$. That is, to find the $x$-th number in the sequence, we can plug in $x$ to $f$.

So we want to find the values of $f(21), f(22), f(23), f(24), f(25)$.

Inspired from hint 2, instead of mapping x to $f(x)$, what if we try to map the relation of $x$ and $f(f(x))$ instead? We get this following pattern.

$x$ $f(x)$ $f(f(x))$ $x$ in binary
$1$ $f(1)=1$ $f(1)=1=1times1$ $1$
$2$ $f(2)=2$ $f(2)=2=1times2$ $10$
$3$ $f(3)=5$ $f(5)=6=2times3$ $11$
$4$ $f(4)=4$ $f(4)=4=1times4$ $100$
$5$ $f(5)=6$ $f(6)=10=2times5$ $101$
$6$ $10$ $12=2times6$ $110$
$7$ $9$ $21=3times7$ $111$
$8$ $8$ $8=1times8$ $1000$
$9$ $21$ $18=2times9$ $1001$

and so on. Based on the provided sequence, it can be said that

That is possibly why $f(f(5))=10=2times5$ but $f(f(7))=21=3times7$.

BONUS:


From rearranging the function we can obtain a new relation

$f(x)=f^{-1}(n_xtimes x)$.

For the values of $x=21,22,23,24,25$, it can be calculated that $n_{21}=3, n_{22}=3, n_{23}=4,n_{24}=2,n_{25}=3$.

Therefore,

Finding $f(24)$ is fun because if we try to construct $x, f(x), f(f(x)), f(f(f(x))), ...$ starting from $x=3$, we get the sequence

So far I can't use $f(22)=f^{-1}(66)$ and $f(25)=f^{-1}(75)$ to my advantage and I'm stuck..

However, I made the connection that the 'root' in title and hint 1 is perhaps meaning 'half-iterate' or 'functional square root', see this Wikipedia page.

Answered by Nuclear03020704 on May 20, 2021

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