asking help in terminology

[franktaw at netscape.net]

Here are a few more examples in the OEIS:
 
A054741, primitive elements not in OEIS.
A013929, primitive elements are A001248.
A046099, primitive elements are A030078.
A091454,  primitive elements are A091455.  I have submitted a correction for A091455; as defined, it should include 210, 330, etc.  I think the actual sequence, as defined by the program, is more interesting than the definition, so I proposed changing the definition.
A033942, primitive elements are A014612.
A096490, primitive elements not in OEIS.
A100716, primitive elements in A051674.
A097603, primitive elements in A000396.
A013590 might include all multiples, this is an interesting question.  If it does, the primitive elements are not in the OEIS.
 
A062457 contains no multiples of other elements, so it is the primitive elements of some sequence.  That sequence is not in the OEIS.
 
(There are a vast number of sequences in the OEIS that consist only of primes, and a fair number that consist only of semiprimes.  Any of these are the primitive elements of some sequence; but enumerating these is uninteresting.)
 
Franklin T. Adams-Watters
 
 
-----Original Message-----
From: franktaw at netscape.net


The primitive abundant numbers are A091191.
 
The composite numbers A002808 have the semiprimes A001358 as primitive elements; there is a comment to this effect.
 
For A009003, the primitive elements are A002144.
 
I would argue against describing this simply as the "kernel" of the sequence.  There are too many kernels of different types floating around already.  "Multiplicative kernel" would be better.  But really, I don't see any problem with saying "primitive elements of" the sequence.  It has the singular advantage that it is already in use.
 
Franklin T. Adams-Watters
 
 
-----Original Message-----
From: hv at crypt.org


A23196 (non-deficient numbers) is another example, which has A6039
("primitive" non-deficient numbers) as its kernel.

A5101 (abundant numbers) also qualifies; its kernel does not appear
to be in OEIS.

Hugo

Emeric Deutsch <deutsch at duke.poly.edu> wrote:
:Dear seqfans,
:
:A question of terminology:
:
:I have a sequence, say A (it does not contain 1). It
:has the property that all multiples of all terms of
:A are in A. What is the best way to describe this?
:Closed under multiplication by positive integers?
:
:Let me call a term of A primitive if no proper divisor
:of A is in A. The sequence S of the primitive terms of A
:is infinite. What is a good term for this subsequence?
:Kernel of A? The prime kernel of A? The primitive elements
:of A? (Clearly, A can be recovered from S.)
:
:I am just afraid that my description of the situation is
:unnecessarily complicated!
:
:A well-known example of this situation is the set
:A={2,3,4,5,...}. In this case S consists of the prime
:numbers.
:
:Do you know of other nontrivial examples? By "trivial"
:example I mean one in which the "prime kernel" is a
:finite set.
:
:Thanks,
:Emeric
:
:P.S. The sequence I have in mind is the sequence in
:my May 16 posting: "a new sequence". The "prime
:kernel" is 6,20,28,45,63,70,... .
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