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<DIV>Here are a few more examples in the OEIS:</DIV>
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<DIV>A054741, primitive elements not in OEIS.</DIV>
<DIV>A013929, primitive elements are A001248.<BR>A046099, primitive elements are A030078.<BR>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.<BR>A033942, primitive elements are A014612.<BR>A096490, primitive elements not in OEIS.<BR>A100716, primitive elements in A051674.<BR>A097603, primitive elements in A000396.<BR>A013590 might include all multiples, this is an interesting question. If it does, the primitive elements are not in the OEIS.</DIV>
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<DIV>A062457 contains no multiples of other elements, so it is the primitive elements of some sequence. That sequence is not in the OEIS.</DIV>
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<DIV>(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.)<BR> </DIV>
<DIV>Franklin T. Adams-Watters<BR> </DIV> <BR>-----Original Message-----<BR>From: franktaw@netscape.net<BR><BR>
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<DIV>The primitive abundant numbers are A091191.</DIV>
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<DIV>The composite numbers A002808 have the semiprimes A001358 as primitive elements; there is a comment to this effect.</DIV>
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<DIV>For A009003, the primitive elements are A002144.</DIV>
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<DIV>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.</DIV>
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<DIV>Franklin T. Adams-Watters<BR> </DIV> <BR>-----Original Message-----<BR>From: hv@crypt.org<BR><BR>
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<DIV class=AOLPlainTextBody id=AOLMsgPart_0_9019c1bb-dc1e-494a-847c-717e91d84473><PRE><TT>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 <<A href="javascript:parent.ComposeTo('deutsch%40duke.poly.edu');">deutsch@duke.poly.edu</A>> 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,... .
</TT></PRE></DIV></DIV></DIV></DIV></DIV></DIV>
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