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The sequence ${\displaystyle{M_n:=2^{p_n}-1}}$, where ${\displaystyle{n\gt0}}$ and ${p_n}$ is the ${\displaystyle{n}^{th}}$ prime number, is commonly known as the Mersenne numbers (not to be confused with the Mersenne primes, which also requires the number itself to be prime). They have the property that no member of this sequence is divisible by ${2}$ (all Mersenne numbers are also odd numbers). In general, the sequences ${\displaystyle{a_{m,n}:=p_m^{p_n}-p_{m-1}\#}}$, where ${\displaystyle{m,n>0}}$ and ${\displaystyle{p_m\#:=\prod_{i=1}^mp_i}}$ is the ${\displaystyle{m^{th}}}$ primorial number, contain no members divisible by the first ${\displaystyle{m}}$ primes. The Mersenne numbers are a special case where ${\displaystyle{m=1}}$.

Do these sequences have a name? Do any important properties of Mersenne numbers besides the indivisibility by the first ${\displaystyle{m}}$ primes generalize to these related sequences?

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  • $\begingroup$ .... and a property of all primes (other than $2$ itself). $\endgroup$ – David G. Stork Aug 24 at 2:55
  • $\begingroup$ @hardmath I did not mean that all odd numbers are Mersenne numbers; I just meant that the Mersenne numbers a subset of the odd numbers (i.e. oddness is a property, but not the defining characteristic, of Mersenne numbers). This was to highlight the fact that the generalized sequences are not divisible by the first m primes. I was wondering if any of the important properties of the Mersenne numbers might also apply to these related sequences. $\endgroup$ – Evan Bailey Aug 24 at 3:15
  • $\begingroup$ @DavidG.Stork I was referring to Mersenne numbers in general, not just Mersenne primes. I apologize if this was not clear in the post; I can edit it if that would be helpful. $\endgroup$ – Evan Bailey Aug 24 at 3:19
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    $\begingroup$ The elements of the sequence $2^{p_n}-1$ are called "Mersenne numbers with prime exponent" , there is no more specific name. The sequences you mentioned in the later part are constructions of numbers having no small prime factor and thus having a "good" chance to be prime. Note that the factors of $2^p-1$ ($p$ prime) are even more restricted : They have to be of the form $2kp+1$ with positive integer $k$. $\endgroup$ – Peter Aug 24 at 12:41
  • $\begingroup$ I think this is an interesting generalization of Mersenne numbers, and I did not not find these sequences described as such in places on the Web like Chris Caldwell's Prime Pages. Some simple factorizations can illustrate whether divisors have properties similar to what @Peter describes. $\endgroup$ – hardmath Aug 25 at 13:50
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Let's broaden the family of sequences somewhat. Let $b,c$ be integers, with $b\gt 1$, and consider the integer sequence for $k = 0,1,2,\ldots$:

$$ s_k = b^k - c $$

The case you ask about is when $b=p$ is prime and $c = (p-1)\#$ is a primorial number, and you restricted attention to the subsequence of prime exponents $k$.

Let's call these exact powers with a fixed offset. Negative as well as positive values of the offset constant $c$ are allowed, so these sequences include as subsequences Fermat numbers as well as Mersenne numbers. Some less well-known sequences in this family are the Cunningham numbers:

$$ b^n \pm 1 $$

and the so-called Crandall primes (after Richard Crandall's U.S. Patent 5,159,632, although there is prior art by Bender and Castagnoli):

$$ 2^q - c \;\;\text{ for small odd } c $$

The latter were studied as providing a richer supply of primes (than the few dozen Mersenne primes) for use as prime moduli of elliptic curve cryptosystems.

The key feature of these sequences is that they result from iteration of a first-degree univariate polynomial:

$$ s_{k+1} = b s_k + (b-1)c $$

This leads to treatment by arithmetic dynamics. I'll illustrate some of the ideas by taking the example $b=3$ and $c=2$ of your constructions.

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