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Published 2002 Revised 2018
This article is about $ 2^n-n $ numbers, that is, numbers that are produced by replacing '$n$' in $ 2^n-n $ with a positive integer $ (1,2,3...) $. I came across these numbers while studying Mersenne numbers $ (2^n-1) $. It got me thinking about $ 2^n-n $ numbers, if there are any interesting properties to them, and what are the properties of their primes. In the rest of the article $A_n$ will
mean $ 2^n-n $. The first few numbers are:
n | A n |
1 | 1 |
2 | 2 |
3 | 5 |
4 | 12 |
5 | 27 |
6 | 58 |
7 | 121 |
8 | 248 |
9 | 503 |
10 | 1014 |
n | 2 n -n | Definite/Probable Prime |
2 | 2 | Definite |
3 | 5 | Definite |
9 | 503 | Definite |
13 | 8179 | Definite |
19 | 524269 | Definite |
21 | 2097131 | Definite |
55 | 36028797018963913 | Definite |
261 | 2 261 -261 | Probable |
3415 | 2 3415 -3415 | Probable |
4185 | 2 4185 -4185 | Probable |
7353 | 2 7353 -7353 | Probable |
12213 | 2 12213 -12213 | Probable |
60975 | 2 60975 -60975 | Probable |
61011 | 2 61011 -61011 | Probable |
for a large $ n $. Taking the harmonic series: $$ \sum_{n=1}^{\infty} \frac{1}{\log(2^n-n)} $$ one will the see that the harmonic series diverges and therefore there are, probably, infinity number of primes of this form.