What is the max of $n$ such that $\sum_{i=1}^n\frac{1}{a_i}=1$ where $2\le a_1\lt a_2\lt\cdots\lt a_n\le 99$?

What is the max of $n$ such that $$\sum_{i=1}^n\frac{1}{a_i}=1$$ where $a_{i}\ (i=1,2,\cdots,n)$ are integers which satisfy $2\le a_1\lt a_2\lt\cdots\lt a_n\le 99$ ?

Also, I need how to prove that the $n$ you get is the maximum.

My approach:

I'm going to represent $\sum_{i=1}^n\frac{1}{a_i}$ as $(a_1,a_2,\cdots,a_n).$

$(2,3,6)\rightarrow(4,5,7,9, 12, 15,18,30,42,45,90)$ $\rightarrow(8,9,10, 12, 14, 15, 16, 18,22,27, 30, 35, 40, 42, 45, 48, 54, 56, 60, 72, 90, 99)$

(here I used $(3)=(4,12), (6)=(7,42)$ etc.)

This is the $n=22$ case.

Update: I've just got the following $n=42$ case. I don't know if this is the max.

$(8,9,10, 12, 14, 15, 16, 18,22,27, 30, 35, 40, 42, 45, 48, 54, 56, 60, 72, 90, 99)\rightarrow(15,17,20,21,22,26,27,30,32,33,34,35,36,38,39,40,42,44,45,48,50,52,54,55,56,60,63,66,70,75,76,77,78,80,84,85,88,90,91,95,96,99)$

Here I used $(10)=(17,34,85), (14)=(28,44,77)$ etc.

Update 2 : I've just got another $n=42$ case.

$(17,18,20,21,22,24,26,27,32,33,34,35,36,38,39,40,42,44,45,48,50,52,54,55,56,60,63,66,70,72,75,76,77,78,80,84,85,88,91,95,96,99)$

Here I used $(15,30,90)=(18,24,72)$.

Update 3 : I crossposted to MO.

https://mathoverflow.net/questions/142300/what-is-the-max-of-n-such-that-sum-i-1n-frac1a-i-1-where-2-le-a-1-l

[EDITED to include other prime powers and give the list of 27 maximal solutions]

The maximum is $42$, attained in $27$ ways (listed below); there are $566$ runners-up with $41$, and then $6747$ with $40$, another $52078$ with $39$, "etc.".

The counts are obtained by dynamical programming. We simplify the computation by checking that the $a_i$ can include no multiple of a prime power greater than $27$, and if multiples of $11$, $13$, $16$, $17$, $19$, $25$, or $27$ appear then they must combine to remove that factor from the denominator, which can only be done in $46,\phantom. 9,\phantom. 7,\phantom., 1, \phantom. 2,\phantom. 1,\phantom. 1$ ways respectively (the last four are $(17,34,85)$, $(19,57,76)$ or $(38,76,95)$, $(50,75)$, and $(25,54)$ respectively; $23$ does not occur). That brings the denominator down to $D = 2^3 3^2 5 \phantom. 7 = 2520$, small enough to make a table of the number of times each pair of integers arises as $(n, D\sum_{i=1}^n 1/a_i)$ with $\sum_i 1/a_i \leq 1$, and at the end extract the counts for $(n,D)$.

This approach does not immediately give the list of $27$ maximal solutions, but it can be modified to compute this list instead: at each stage, instead of recording the number of representations of each fraction, keep track of the representation(s) with the largest number of terms. The list of $42$'s is follows, in lexicographical order; each uses the prime 17, and all use 99 except for two which have $\max_i a_i = 96$.

[12, 17, 21, 22, 24, 26, 27, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 48, 50, 52, 54, 55, 56, 60, 63, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[13, 17, 18, 21, 22, 24, 26, 27, 32, 33, 34, 35, 38, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 60, 63, 65, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[13, 17, 18, 22, 24, 26, 27, 28, 32, 33, 34, 35, 36, 38, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 60, 65, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[13, 17, 20, 21, 22, 24, 26, 27, 32, 33, 34, 35, 36, 38, 40, 42, 44, 48, 50, 52, 54, 55, 56, 60, 63, 65, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[13, 17, 20, 21, 22, 24, 26, 30, 32, 33, 34, 35, 36, 38, 40, 42, 44, 45, 48, 50, 52, 55, 56, 60, 63, 65, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[13, 17, 20, 21, 22, 26, 27, 30, 32, 33, 34, 35, 36, 38, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 60, 63, 65, 66, 70, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[13, 17, 20, 21, 24, 26, 27, 30, 32, 33, 34, 35, 36, 38, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 60, 63, 65, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96]
[13, 17, 20, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 38, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 60, 65, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[14, 17, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 44, 48, 50, 52, 54, 55, 56, 60, 63, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[15, 17, 18, 21, 22, 24, 26, 27, 32, 33, 34, 35, 38, 39, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 60, 63, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[15, 17, 18, 22, 24, 26, 27, 28, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 60, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[15, 17, 20, 21, 22, 24, 26, 27, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 48, 50, 52, 54, 55, 56, 60, 63, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[15, 17, 20, 21, 22, 24, 26, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 48, 50, 52, 55, 56, 60, 63, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[15, 17, 20, 21, 22, 26, 27, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 60, 63, 66, 70, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[15, 17, 20, 21, 24, 26, 27, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 60, 63, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96]
[15, 17, 20, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 38, 39, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 60, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[17, 18, 19, 21, 22, 24, 26, 27, 30, 32, 33, 34, 35, 39, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 57, 60, 63, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 96, 99]
[17, 18, 19, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 39, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 57, 60, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 96, 99]
[17, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 38, 39, 40, 44, 45, 48, 50, 52, 54, 55, 56, 60, 63, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[17, 18, 20, 21, 22, 24, 26, 27, 30, 32, 33, 34, 35, 38, 39, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 63, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[17, 18, 20, 21, 22, 24, 26, 27, 30, 32, 33, 34, 35, 38, 39, 40, 42, 44, 45, 48, 50, 54, 55, 56, 60, 63, 65, 66, 70, 72, 75, 76, 77, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[17, 18, 20, 21, 22, 24, 26, 27, 30, 32, 33, 34, 36, 38, 39, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 60, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[17, 18, 20, 21, 22, 24, 26, 27, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 60, 63, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 91, 95, 96, 99]
[17, 18, 20, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[17, 18, 20, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 48, 50, 54, 55, 56, 60, 65, 66, 70, 72, 75, 76, 77, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[17, 18, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 60, 66, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99]
[17, 19, 20, 21, 22, 24, 26, 27, 30, 32, 33, 34, 35, 36, 39, 40, 42, 44, 48, 50, 52, 54, 55, 56, 57, 60, 63, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 96, 99]

Your $22$ is not the max because $(12)=(19,57,95)$. And after that $(9)=(12,36)$.