Today I came across an interesting sequence at OEIS, A068374, described as "Primes $n$ such that positive values of $n$-Primorial($k$) are all primes ($k\gt0$)".

The sequence is as follows:

$(2, 5, 13, 19, 43, 73, 103, 109, 229, 313, 883, 1093, 1489, 1699, 1789, 2143, 3463, 3853, 5653, 15649, 21523, 43789, 47743, 50053, 51199, 59473, 86293, 88819, 93493, 101533, 176053, 197299, 205663, 235009, 257503, 296509, 325543, 338413, 347989)$

I understand that these are the first elements, as long as the calculation of the primorials does not take very long. There is not much information about it, only the user that did it and the basic description. E.g. a PARI program to obtain the sequence is not included (I am editing now the sequence to add some information and will make some PARI calculations), but I did a quick check with Python and observed that all the elements are part of a couple of twin primes, except the first element (2).

Please, I would to ask the following questions:

If the sequence is including only odd twin primes, would demonstrating that it is an infinite sequence (or not) be equivalent to the twin prime conjecture?

Anyway I guess that the difficulty is the same one, but finding a sequence based only on twin primes and related to primorials, seems an interesting point. Thank you!

  • 1
    $\begingroup$ Interesting. It appears also that the numbers always resemble the larger of the primes in the pair. $\endgroup$
    – Daniel R
    Dec 14, 2015 at 8:46

1 Answer 1


Since Primorial$(1)$ is $2$, any prime $n\gt 2$ in this sequence must by definition have $n-2$ being prime, and so $n$ must be the higher of a pair of twin primes.

So if this sequence is infinite then there must be an infinite number of twin primes.

However, I would have thought the reverse is not necessarily true, since there are twin primes which do not appear in this sequence. For example $7$ and $31$ and $61$ are not in this sequence despite being prime and $2$ more than a prime. So you could (and probably do) have an infinite number of twin primes not in this sequence, possibly without this sequence being infinite.

  • $\begingroup$ thank you for the confirmation! $\endgroup$
    – iadvd
    Dec 14, 2015 at 8:50
  • $\begingroup$ Consider tightening up the last paragraph up by saying there are twin prime pairs for which the higher of the twins does not appear in the sequence. $\endgroup$ Jan 25, 2023 at 11:40
  • $\begingroup$ @samerivertwice happier now? $\endgroup$
    – Henry
    Jan 25, 2023 at 12:28

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