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4 added 132 characters in body

The set $S$ of even perfect numbers $n$ such that $n+1$ is a prime number contains $$6,28,33550336,137438691328$$

Latter number found by Joerg Arndt, corresponds to $M_{19}$ (mersenne)

Question: Is $S$ reduced to these $4$ numbers.

New: Joerg Arndt checked up to exponent $110503$ that the corresponding number $n+1$ is composite. (Improved $19$ to $110503$).

Which function of $x$ migh describe well the size of the set of elements in $S$ less than $x$

divided

by the size of the set of all even perfect numbers less than $x$; mainly with big $x.$

So, I am asking for relative size not absolute size. E.g., if I were asking for relative density of the prime numbers congruent to $3$ modulo $4$: I do not want to use the big machinery of the prime number theorem, or Dirichlet's Theorem to deduce how many should be there. I just want (in these case) to know how to describe in terms of $x$

number of primes congruent to $3$ modulo $4$ and less than $x$

divided by

number of primes less than $x$

How many such numbers $n$ we may expect inside the known 47 perfect numbers ?

3 added 80 characters in body

The set $S$ of even perfect numbers $n$ such that $n+1$ is a prime number contains $$6,28,33550336,137438691328$$

Latter number found by Joerg Arndt, corresponds to $M_{19}$ (mersenne)

Question: Is $S$ reduced to these $4$ numbers.

Which function of $x$ migh describe well the size of the set of elements in $S$ less than $x$

divided

by the size of the set of all even perfect numbers less than $x$; mainly with big $x.$

So, I am asking for relative size not absolute size. E.g., if I were asking for relative density of the prime numbers congruent to $3$ modulo $4$: I do not want to use the big machinery of the prime number theorem, or Dirichlet's Theorem to deduce how many should be there. I just want (in these case) to know how to describe in terms of $x$

number of primes congruent to $3$ modulo $4$ and less than $x$

divided by

number of primes less than $x$

How many such numbers $n$ we may expect inside the known 47 perfect numbers ?

2 added 566 characters in body

The set $S$ of even perfect numbers $n$ such that $n+1$ is a prime number contains $$6,28,33550336 6,28,33550336,137438691328$$

Latter number found by Joerg Arndt, corresponds to $M_{19}$ (mersenne)

Question: Is $S$ reduced to these $3$ 4$numbers. Which function of$x$migh describe well the size of the set of elements in$S$less than$x$divided by the size of the set of all even perfect numbers less than$x$; mainly with big$x.$So, I am asking for relative size not absolute size. E.g., if I were asking for relative density of the prime numbers congruent to$3$modulo$4$: I do not want to use the big machinery of the prime number theorem, or Dirichlet's Theorem to deduce how many should be there. I just want (in these case) to know how to describe in terms of$x$number of primes congruent to$3$modulo$4$and less than$x$divided by number of primes less than$x\$

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