# [seqfan] Re: A generalization of "antipodes": primes and highly composite numbers

jean-paul allouche jean-paul.allouche at imj-prg.fr
Fri Jul 22 16:53:36 CEST 2016

```Dear Vladimir

I haven't really thought of your sequences,
but it suddenly vaguely (very vaguely) reminded
me something I have read long ago. I won't be
able to look at that right now, but ---related or not---
you might be interested in looking at
http://arxiv.org/abs/0711.0865

best wishes
jean-paul

Le 22/07/16 à 10:37, Vladimir Shevelev a écrit :
> Dear SeqFans,
>
> I with Peter submitted 12 sequences:
> A275246, 248,249,251,252,253 and
> A275239, 240,241,242,243,244 .
>
> The first series is devoted to the following
> generalization of primes:
> The sieve of Eratosthenes removes all
>   numbers from a list of primes p divisible
>   by p as p is encountered in the list.
> We denote by E* an Eratosthenes-like
> algorithm in which we remove for number
> m all numbers n for which GCD(n,m)>1.
> Removing 1 and primes from the positive
> numbers and using for the
> remaining set E*, we find the smallest
> sequence of pairwise relatively prime
> numbers of kind 1, consisting of the
> squares of all primes (A001248).
> Removing 1, primes, and squares of
> primes from the positive numbers in the
> same way we find the
> smallest sequence of pairwise relatively
> prime numbers of kind 2 (A089581), etc.
>
> Note that the smallest sequence of
> pairwise relatively prime numbers of kind k>0
> begins with 2*(k+1) (which is a unique even
> number in the sequence).
>
> The second series in the similar manner
> is devoted to a generalization of highly
> composite numbers (A002182).
> Let us name them by highly composite
> numbers of kind 0. Removing A002182
> from the positive integers, we build for
> the remaining set, by the rule of A002182,
> so-called highly composite numbers of kind 1.
>   Removing A002182 and the highly composite
> numbers of kind 1 from the positive integers,
> we are building for the remaining set, by the
> rule of A002182, so-called highly composite
> numbers of kind 2, etc.
>
> Note that the sequence of highly composite
> numbers of kind h>=1 begins from Prime(h+1)
> (which is a unique prime in this sequence).
>
> Did anyone hear about such or closed type of
> generalizations?
>
> Best regards,