There is proof that there is no such whole number.
As mentioned in the previous thread: For sufficiently large n there is proof that the density decreases, so the density primes in 2…n can’t be smaller than the same for 2…2n
And for smaller n we can just try all n’s:
In the free version I get that to show the ratio from 2-160, the closest it gets is 1.11 at n=36
And 160 is sufficiently large to be a sufficiently large n for the previously mentioned purpose.
Since the question has been completely answered, I will just throw in this ditty that the professor in a number theory course I took over 60 years ago:
Chebycheff proved it/ You can too/ There’s always a prime/ 'Tween x and x times 2". This is in fact true for x >1; x doesn’t have to be an integer.