When trying to obtain the value of Jacobsthal's function for some $n$; to find the largest sequence of consecutive numbers that are all coprime to $n$, one approach (and the only direct approach that I know of) is to exhaust all of the possible sequences of consecutive numbers that are all coprime to $n$ by trial. But when doing this, it's helpful (when working by hand) to represent all integers (except $1, 0$ and $-1$) by their lowest prime divisor. For example; the sequence $2,3,4,5,6,7,8,9,10$ would be equivalent to $2,3,2,5,2,7,2,3,2$. This representation has its advantages as it represents more than one sequence of consecutive integers. For example; $2,3,2,5,2,7,2,3,2$ is also the respective representation of $212,213,214,215,216,217,218,219,220$. This representation enables the exhaustive approach for determining the value of Jacobsthal's function for some $n$.

But the thing is, I can't find a specific name for this representation of integers, (or this type of sequence in a more context-free setting). Is there a referable/favoured name for this representation? Usually, I have to make up my own dummy name for them to be able to say anything about them, which I would like to avoid if possible.

For example; I would like to conjecture that for any unknown name of the first $n$ prime numbers, that is larger than $2p_{n-1} -1$, there exist a reflection point in the unknown name of some prime numbers, whom when multiplied together are larger than ${p_n}^2$. This is just another way of conjecturing that a prime number exist in all intervals of length $2p_{n-1}$, bounded above by ${p_n}^2$. That's an open question, but I would like to read more about this sort of approach, and related attempts to identify how these unknown name are distributed in respect to primorial numbers, and squared primes.

  • 1
    $\begingroup$ I recommend an acronym to represent the key property. Since it appears you are using the smallest primes, you have an LPF representation, where the member of the sequence corresponding to n is LeastPrimeFactor(n). You can modify this given a base m to change smallest set to set of primes dividing m, and then use LPFM(n) or something similar. If you define it carefully and clearly, using acronyms should not detract much from the presentation. Gerhard "FWIW YMMV IMHO IANAL TISMFISDIMO" Paseman, 2018.04.18. $\endgroup$ – Gerhard Paseman Apr 18 '18 at 15:57
  • $\begingroup$ "LPF sequence" could work... $\endgroup$ – Brad Graham Apr 18 '18 at 16:12
  • 2
    $\begingroup$ Except using the variable name $m$ as part of the acronym (as in LPFM for "least prime factor dividing $m$") tweaks at least my sense of æsthetics in an unpleasant way …. $\endgroup$ – LSpice Apr 18 '18 at 17:01
  • $\begingroup$ You could refer to it by its OEIS id: oeis.org/A020639 $\endgroup$ – José Hdz. Stgo. Apr 19 '18 at 1:02

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.