[seqfan] Re: Prime signature of 1, and second signature

[Matthew Vandermast]

(It seemed advisable to cut some messages...)

Charles Greathouse wrote:

>I guess you could look at it that way: the prime signature is an
ordered finite collection (i.e., a tuple) and there are different
types of prime signatures, the "OEIS prime signature" and the
"MathWorld prime signature" which are the reverses of each other.  I
prefer to think of it as the multiset, and consider the OEIS and
MathWorld ways as just two different conventions on writing the
multiset.  I can't see any reason to do otherwise, but YMMV.

To be honest: Personally, I'd prefer not to think about the nondecreasing version at all if I can help it.  I'm used to the traditional OEIS version, and I like that the nonincreasing version corresponds with A025487. But I respect that others' mileage will vary there.

A question in which I'm more interested is: 
Has it been a unanimous understanding that the prime signature is the multiset?  
I'm still respectfully wondering whether it's possible that, from early on, some people understood the prime signature to be the multiset itself, and others understood it to be something else (such as a list) that *signified* that multiset.  I'd still be interested to see pre-2006 definitions of the prime signature in print or online.

I've noted that MathWorld defines prime signature as a list, not a multiset.  Interestingly, MathWorld uses "braces" or "curly brackets" ( { } ) for prime signatures, and these are more characteristic of multisets than of lists.  On the other hand,  the familiar OEIS comment that I quoted before uses parentheses ((  )), and those are more characteristic of a list than a multiset.

Anyway, based on this sample of responses, the current majority position certainly seems to be "prime signature=multiset."  I plan to write my prime signature sequence (A217171) so that it's compatible with either "signature=multiset" or "signature=identifier of that multiset." (And I'll leave out the 0 for n=1.)   I'll try to define A212172 so that it's as ecumenical as possible, too, but I'd still like to use 0 for squarefree n. Except for the prime signature sequence, it seems that the OEIS usually tries to have an a(n) for as many values of n as possible, and leaving a(n) unenterable for over 60% of n still seems unnecessary to me. Let's see how it goes. Thanks again for everyone's input (and please feel free to add more). 

Regards,
Matt Vandermast 

----- Original Message -----
From: Charles Greathouse <charles.greathouse at case.edu>
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Sent: Mon, 11 Jun 2012 18:58:38 -0000 (UTC)
Subject: [seqfan] Re: Prime signature of 1, and second signature

I guess you could look at it that way: the prime signature is an
ordered finite collection (i.e., a tuple) and there are different
types of prime signatures, the "OEIS prime signature" and the
"MathWorld prime signature" which are the reverses of each other.  I
prefer to think of it as the multiset, and consider the OEIS and
MathWorld ways as just two different conventions on writing the
multiset.  I can't see any reason to do otherwise, but YMMV.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University


On Mon, Jun 11, 2012 at 12:56 PM, Matthew Vandermast
wrote:
>>I'm not sure what you mean by a list as distinct from a multiset.
> Normally I would take this to mean a tuple, but that can't be the
> right definition here since you want the signature of 18 to be the
> same as the signature of 12.
>
> Let me address this question as best I can starting with MathWorld.  MathWorld defines n's prime signature (when n >1) as a "sorted list of nonzero exponents."  MathWorld (and A118914) sorts these exponents in nondecreasing order. In that version of prime signature, the prime signature of both 12 and 18 is the list (1,2). The exponents could also be sorted (and in the OEIS, I believe they conventionally have been sorted) in nonincreasing order.  In that version, the prime signature of both 12 and 18 is the list (2,1).
>
> It's my understanding that the lists (1,2) and (2,1) qualify as different lists (even though 12 and 18 correspond to the same list in each version of prime signature).  But the two seemingly-different lists express the same multiset.  Therefore, it's my understanding that a list is, in principle, conceptually different from the multiset it represents. (Note: This doesn't depend on whether one accepts or rejects MathWorld's definition of a prime signature as a list and not a multiset.)  If this is an error, I hope it's at least an understandable one.
>
> I'm not sure I feel up to defining the concept "list" at the moment, but it's my understanding that the makeup of a specific list depends not only on the multiset it represents, but the way that multiset is ordered in the list.  I could be wrong about this. On the other hand, perhaps sources (if any) that imply otherwise could be wrong.  On the third hand, maybe it's an unsettled terminological question at this point.
>
> Thanks,
> Matt Vandermast
>
>
>
> ----- Original Message -----
> From: Charles Greathouse 
> To: Sequence Fanatics Discussion list 
> Sent: Mon, 11 Jun 2012 15:30:02 -0000 (UTC)
> Subject: [seqfan] Re: Prime signature of 1, and second signature
>
>> I don't want to base too much on MathWorld's prime signature page (as great as that site usually is). But what I'm respectfully wondering at the moment is:  Is it possible that, from early on, there were variant understandings of what prime signature meant?  Perhaps some people understood the prime signature to be the multiset itself, and others understood it to be something (a list of exponents, when n > 1) that *signified* that multiset?
>
> I'm not sure what you mean by a list as distinct from a multiset.
> Normally I would take this to mean a tuple, but that can't be the
> right definition here since you want the signature of 18 to be the
> same as the signature of 12.
>
> Charles Greathouse
> Analyst/Programmer
> Case Western Reserve University