Let's take a sequence that is the sum of the reciprocals of consecutive prime numbers. To check its consistency or to contradict it, I created an Excel spreadsheet which can be downloaded here.

Of course, a spreadsheet like this has a finite range of numbers to check, so I made a mistake and I have to admit it.

First, I examined the sum of the reciprocals of the next 1,024 prime numbers and began to have my suspicions. To verify them, I ambitiously increased the range to 10,000 numbers and... I was happy because it looked like my suspicions were justified:

The sum of the reciprocals calculated in this way was 2.70925824879732

and differed from 𝒆 only by 3.32 ‰ (per mille). Ha, I told myself, it can't be a coincidence, there are NO such cases!

However, a discussion forum consultation gave me an informed voice, referring me to Wolfram's website, where this series was calculated for 50,000 prime numbers, and the result dashed my hopes.

Since I couldn't find any details in Wolfram, like unfaithful Thomas, I started expanding the scope of my spreadsheet, but stopped at 20,000 primes. This turned out to be too ambitious a move, as the Euler number was exceeded by the sum of the reciprocals of 10,998 numbers. So with the prime number 116437

Because I did a solid job that no one needed, I am sharing its result, i.e. the mentioned sheet. Contains 20480 NUMERICAL (very useful!) prime numbers, the highest is 230467

Line 3 contains their inverses, which are carefully added in line 4.

Every tenth column of line 5 shows these values with more digits so you can follow the progress of the calculation.

Line 7 gives the averaged (64 numbers) local distances between prime numbers, i.e. their local frequency. This is an interesting calculation because it shows something that is a bit surprising to me: these distances do not increase significantly with the size of the numbers! Even after crossing the 10,000 mark, approximately one in twelve numbers is a prime.

Line 9 gives this proportion for the full range, i.e. starting from the digit 2.

On sheet 44, line 11, starting at position 10982 (HV), shows how the Euler number is exceeded. The given values should be divided by one million.

I am presenting the results of my work, hoping that they will prove useful for examining interesting regularities or other features occurring among prime numbers.

@BaSzRafael on 𝕏