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

Matthew Vandermast ghodges14 at comcast.net
Sun Jun 10 21:06:23 CEST 2012

Thanks, David.  I've been curious about the history of the phrase "prime signature." Did you invent the phrase and/or first define the concept? If so, my hat is off to you (if not, my hat is still off just for submitting the sequence, which is perhaps my favorite in the database).  I was also influenced by your A001055 comment when writing my original e-mail about this.  

Unless there are specific drawbacks to this, I'd like A212172 to "signify" n's multiset of exponents >=2 by listing the exponents in nonincreasing order (i.e., their natural order in the prime factorization of a member of A025487)  when the multiset in question is nonempty, and with a 0 when it's the empty multiset.  My  definition of A212172 seems pretty normal (to me, at least) for the OEIS. 

Again, I think the relevant definition of "signature" is "an identifying characteristic or mark." 
With the prime signature, when only 1 has the empty multiset of positive exponents, it's not such a problem whether one can identify that multiset in the sequence (although there may or may not be positives to this, depending on one's point of view; I personally see some). It can be done in a line on the sequence page.
With the second signature, the issues are different. A signature that can't be entered for over 60% of n isn't much of an identifier at all.

It's not just A212172 I'm concerned about; I also like being able to signify the second signature of the primorials in A212175  (https://oeis.org/draft/A212175), and I'm sure there are, or will be, other ramifications.


----- Original Message -----
From: David Wilson <davidwwilson at comcast.net>
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Sent: Sun, 10 Jun 2012 18:05:24 -0000 (UTC)
Subject: [seqfan] Re: Prime signature of 1, and second signature

Having authored A025487, I would beg to differ.

The prime signature of n is the multiset of positive exponents occurring 
in the prime factorization of n.

The prime signature of 1 is the empty multiset, since 1 has no prime 

Given the representation of multisets in A118914, it is correct that 
there are no terms associated with 1.

IMHO, 0 is not an acceptable substitute for the prime signature of 1. By 
the definition above, a prime signature is a multiset of positive 
integers, hence 0 cannot be an element of a prime signature.

Of course, when you submit a sequence, you can include whatever kludge 
you wish to represent 1, it is my observation that it is better to omit 
elements entirely than kludge them. Better to start at index 2 than to 
treat 1 inconsistently.

On 6/10/2012 7:14 AM, Matthew Vandermast wrote:
> Thanks, Charles.  Just to clarify the relevant part of a long email, I'm suggesting that a "signature" is distinguishable from a "multiset," and that 0 be used as the signature of the empty multiset in those sequences. I hope my suggestion will be acceptable, and it seems consistent with OEIS conventions to me.


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