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

Sat Jul 23 17:38:50 CEST 2016

```Dear Jean-Paul,

>...vaguely (very vaguely) reminded
>me something I have read long ago

Let me to remark that often simple
but interesting things seem to be
by Olivier).
Maybe, you mean a similar way, using
a standard Eratosthenes algorithm,
when we consistently obtain primes,
semiprimes, numbers with 3 (maybe,
equal) prime factors, etc. (although
I did see nothing). This case
seems less interesting. Maybe,
you saw something even
before the foundation of OEIS, otherwise
these sequences more definitely
would be placed in OEIS. In any case
it is interesting what have you read
on this topic.

A separate thanks for a paper.

Best regards,

________________________________________
From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of jean-paul allouche [jean-paul.allouche at imj-prg.fr]
Sent: 22 July 2016 17:53
To: seqfan at list.seqfan.eu
Subject: [seqfan] Re: A generalization of "antipodes": primes and highly composite numbers

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,
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/

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