I am hoping someone can estimate the number of primes that come up this way: take a number $L,$ then let $$ C = \operatorname{lcm} (1,2,3,\ldots,L). $$ We know that $C$ has quite a lot of divisors; indeed, this is just a variant of Ramanujan's Superior Highly Composite Numbers, in that the exponent of some prime $p$ is roughly proportional to $1/\log p.$ Indeed, for this case, the exponent is $$ \left\lfloor \frac{\log L}{\log p} \right\rfloor \; \; .$$
For every divisor $d |C,$ let $q = 1 + d.$ I would like an estimate of the number of $q$ that come up prime. For example, with $L=4,$ we get $C=12$ and divisors $1,2,3,4,6,12,$ thence primes $2,3,5,7,13.$ Below is a table for $L \leq 61;$ if $L$ is as above, and $P$ is the count of primes, the number ``log ratio'' is $(\log P) / (\log L).$ This and other calculations suggest that $P$ is just short of exponential in $L,$ just a little slower growth than $e^{L / \log L}.$
lcm of up to: 4 divisors : 6 prime count: 5 log ratio: 1.160964047443681
lcm of up to: 5 divisors : 12 prime count: 8 log ratio: 1.292029674220179
lcm of up to: 7 divisors : 24 prime count: 13 log ratio: 1.318123223061841
lcm of up to: 8 divisors : 32 prime count: 15 log ratio: 1.30229686520284
lcm of up to: 9 divisors : 48 prime count: 22 log ratio: 1.406794046107798
lcm of up to: 11 divisors : 96 prime count: 36 log ratio: 1.494443472618428
lcm of up to: 13 divisors : 192 prime count: 66 log ratio: 1.633425911446773
lcm of up to: 16 divisors : 240 prime count: 81 log ratio: 1.584962500721156
lcm of up to: 17 divisors : 480 prime count: 148 log ratio: 1.763796674277192
lcm of up to: 19 divisors : 960 prime count: 252 log ratio: 1.877922798412615
lcm of up to: 23 divisors : 1920 prime count: 446 log ratio: 1.945568555358454
lcm of up to: 25 divisors : 2880 prime count: 660 log ratio: 2.016927706518875
lcm of up to: 27 divisors : 3840 prime count: 905 log ratio: 2.065616479359747
lcm of up to: 29 divisors : 7680 prime count: 1638 log ratio: 2.197974766132367
lcm of up to: 31 divisors : 15360 prime count: 2912 log ratio: 2.322837836698491
lcm of up to: 32 divisors : 18432 prime count: 3578 log ratio: 2.360987534410382
lcm of up to: 37 divisors : 36864 prime count: 6661 log ratio: 2.438168109993427
lcm of up to: 41 divisors : 73728 prime count: 12344 log ratio: 2.536890418395409
lcm of up to: 43 divisors : 147456 prime count: 23060 log ratio: 2.670917389347842
lcm of up to: 47 divisors : 294912 prime count: 42735 log ratio: 2.769445392395919
lcm of up to: 49 divisors : 442368 prime count: 63329 log ratio: 2.840855381850106
lcm of up to: 53 divisors : 884736 prime count: 118262 log ratio: 2.942014853177503
lcm of up to: 59 divisors : 1769472 prime count: 221800 log ratio: 3.018864087268213
lcm of up to: 61 divisors : 3538944 prime count: 417192 log ratio: 3.148065898896646
So that's the question, can anyone give a reasonable estimate on $P,$ possibly conditional on open conjectures? Oh, not by the way, what I really want to know is closer to the product of all those primes; in case of interest, see https://math.stackexchange.com/questions/1334767/the-maximal-size-of-between-varphin-divided-by-lambdan/1336917#1336917
