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

[Marc LeBrun]

Please avoid abbreviations, they make searching difficult for those that don't know your secret jargon.

In names, titles and short comments, just go ahead and say “highly composite”.  

In a longer comment or explanation, always introduce the abbreviation before using it: 
“…numbers of highly composite (HC) class 1… numbers of HC class 2…” etc.

 It’s not like the Internet is going to run out of bits any time soon.


> On Jul 22, 2016, at 6:17 AM, Neil Sloane <njasloane at gmail.com> wrote:
> 
> 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.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> 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,
>> Vladimir
>> 
>> --
>> Seqfan Mailing list - http://list.seqfan.eu/
>> 
> 
> --
> Seqfan Mailing list - http://list.seqfan.eu/