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

Neil Sloane njasloane at gmail.com
Fri Jul 22 15:17:22 CEST 2016

```Vladimir,

I tried to say this in a pink box comment to A275239, but the OEIS too slow
today

"kind 1" etc is not good English.

I suggest either "type 1" or "class 1". However, these are very common
names.

But even better, how about "of HC class 1" (where HC stands for "Highly
composite")?

"Numbers of HC class 0" ,
"Numbers of HC class 1" ,
etc

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Email: njasloane at gmail.com

On Fri, Jul 22, 2016 at 4:37 AM, Vladimir Shevelev <shevelev at bgu.ac.il>
wrote:

> 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,