The Security to the Party [Part 41]

Question

After many, many years from the last spaghetti party, the host of the previous 40 parties has created a discord server named OEIS (the name of the company) open to the people that still hung around with the same email address with the ingredients of spaghetti recipe (and not the steps of making it) posted in a secret channel, and you and your friend (Sam) have another friend, Tom, a discord moderator of the server. Tom, being an average guy, just gives you and Sam the invite link to the discord server, but does not tell you how you should figure the code out. However, he says that you need an accurate calculator for hard codes.

You and Sam enter the discord server together, and see the chat history for the past day. There has been five people that have attempted to get in the server, and it seems like a very simple code to Sam.

The discord moderator said to the first person "1", and the person says "1" and is let in.
The discord moderator said to the second person "4", and the person says "4" and is let in.
The discord moderator said to the third person "10", and the person says "10" and is let in.
The discord moderator said to the fourth person "28", and the person says "19" and is let in.

Sam now tries to get in to the locked channels in the server because he seems fairly confident. The discord moderator said to him "16", and Sam says "16" but he is banned from the server.

The discord moderator now says "34" to you, so what should you say? (There are at least two solutions.)

Bonus Question

There is a 7-digit passcode (capital letter and 6 numbers) access to a pdf to the steps of making the spaghetti. Can you find clues in the question for that?

Hints for Second Solution:

Hint 1:

Some very powerful puzzle. (Watch for puns)

Other Puzzles In This Series:

No. 1 - Closedㅤㅤ No. 11
No. 2ㅤ ㅤ ㅤ ㅤ ㅤ $\,$No. 12
No. 3 - Closedㅤ$\ \,$ No. 13
No. 4
No. 5
No. 6
No. 7
No. 8 - Closed
No. 9
No. 10

Answer 1

You should say:

22 and as a reply to 16, you should have said 13.

The answer ...

... can be found as follows: Consider the string S of all positive integers concatenated. Now find the position P of an integer N in S. The moderator gives you the position P and your reply should be N. (The moderator must take care to use only numbers that occur in the sequence P.)

 ····|····1····|····2····|····3····|····4····|···
 123456789101112131415161718192021222324252627...
 ^  ^     ^^    ^^          ^^    ^^
 1  4     10    16          28    34

The PDF with the recipe is protected ...

... with the password A117804, because that's the id of that sequence on OEIS, the online encyclopedia of integer sequences, which has a useful search facility. (The answer requires a reverse look-up: The input is a number in that sequence and the answer is its index.)

Answer 2

Here is the second solution to my own problem.

You should say:

25, and your friend should have said 7

The hint, which says that this is Some very powerful puzzle, actually means

Sum (of digits of) power(s) (of) four

We can also compare this to the sequence of

$3n+1$

And compare these two sequences like this (Don't scroll down if you don't want to as the table cannot be spoilered):

$n$	$3n+1$ (What they ask you)	$4^n$	Sum of Digits of $4^n$ (What you answer)
0	1	1	1
1	4	4	4
3	10	64	10
5	16	1,024	7
9	28	262,144	19
11	34	4,194,304	25

The pdf to the recipe is locked with the code:

A065713, with the sequence (reference by the company name) here: http://oeis.org/A065713