[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]
Best Answer
We assume $a_1 < a_2 < \dots$.
It is easy to get an answer with $n = 4$: $\frac 1 2 + \frac 1 3 + \frac 1 {12} + \frac 1 {156}$.
To see that $n = 2$ doesn't work, just note that $\frac 1 2 + \frac 1 3 = \frac 5 6 < \frac{12}{13}$.
To see that $n = 3$ doesn't work:
Assume there is a solution. Note that $\frac 1 3 + \frac 1 4 + \frac 1 5 = \frac{47}{60} < \frac{12}{13}$, so we must have $a_1 = 2$. This leads to $\frac 1 {a_2} + \frac 1{a_3} = \frac{11}{26}$.
Therefore $\frac 2 {a_2} > \frac {11}{26}$ and hence $a_2 \leq 4$. Checking $a_2 = 3$ and $a_2 = 4$, we see that $a_3$ is not an integer in each case.