Consider the series : $1,2,4,8,16,23,28,38,49,\ldots $
( number + sum of its digits numbers gives the next number and continues up to infinity )
Now, consider the primes in this series ( the first ones $2,23,101,103,107 , \ldots $)
Now, can you help with the following questions :
1) what is the reciprocal sum of primes in this series? ( my guess is around 0.7 which makes the series converges but how I can prove that ? )
2) How many consecutive primes can be found in this series? ( My guess is 7 or 8 , but still I am not able to prove this result, for example, 101,103,107 are the consecutive primes in the series )
3) It is believed that $\frac{x}{(\ln x)^2}$ roughly gives the number of twin primes and I suspect that that formula works for the number of primes in the series ( $1,2,4,8,16,23,28,38,\ldots$ )
Any help will be appreciated. Thank you.