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

Fri Jul 22 10:37:43 CEST 2016

```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,