[seqfan] Prime signature of 1, and second signature

Matthew Vandermast ghodges14 at comcast.net
Sun Jun 10 04:15:34 CEST 2012

Hello Seqfans,

A118914 (https://oeis.org/A118914) currently has no value for n = 1; the reasoning is explained on the sequence page.  First: I agree wholeheartedly with Daniel Forgues that A118914(1) shouldn't have been 1. This implied that 1 and 2 had the same prime signature, an implication that contradicted A025487 and many related sequences.  I also thank both Forgues and MathWorld for helping me appreciate the distinction between "set" and "multiset" (both of which are distinct from "list").  

But just as the concepts "set," "multiset," and "list" are mutually distinct, each of those concepts can also be distinguished from the concept "signature" (although in the right circumstances, lists and other entities can serve perfectly well as signatures).  My dictionary defines "signature" as "an identifying characteristic or mark." Usually, it's natural and conventional to identify the multiset of multiplicities of n's prime factors with an ordered list of the associated positive exponents. When n =1, however, this multiset is { } (the empty set or multiset). 

It's not obvious how to "signify" { } in the OEIS. But any such "signature" that can be entered in an actual OEIS sequence must be an integer or collection of integers. It's my understanding (subject to correction by Neil, obviously!) that, essentially, the normal OEIS custom is to signify { } with a 0 unless other considerations make that impossible. For example, there may be other 0s in the sequence.  (Sequences such as A001055 and A005361 represent the *product* of the empty multiset as 1; they do not signify { } itself as 1.)  

Certainly, hundreds of OEIS sequences (e.g., https://oeis.org/A003321) are of the general form: "Smallest (largest, etc.) number related in such-and-such a way to n, or 0 if no such number exists." This seems equivalent to: When the set of numbers with the defined relationship to a specific n is the empty set, a 0 is used to signify that set. In any case, amending the definition of A118914 to include the phrase "or 0 if no such exponent exists," thus making A118914(1) = 0, would seem to fit very well with general OEIS conventions, including the convention of starting a sequence as early as reasonably possible.

Here's the biggest reason I care about this:
I'll be submitting A212172 (https://oeis.org/draft/A212172) shortly, which gives the "second signature" of n. For non-squarefree n, this will be an ordered list of exponents >=2 in the canonical prime factorization of n.  Since the multiset of exponents >=2 for squarefree n is { }, my definition includes the phrase "or 0 if no such exponent exists."   I hope this will be acceptable.  The second signature has many interesting properties (see draft) and I like it a lot; there seems no reason to give it the "ugly duckling" treatment by defining it so that the signature cannot be entered in the OEIS for over 60% of n, when there's a conventional and logical alternative.  

I'll also be submitting A212171 (https://oeis.org/draft/A212171) shortly, which defines the "nonincreasing version" of n's prime signature as the "list of positive exponents in [the] canonical prime factorization of n, in nonincreasing order, or 0 if no such exponent exists." This seems to fit the same OEIS conventions that an analogous definition for A118194 would fit. (Incidentally, the practice of listing the exponents in nonincreasing order -  i.e., their natural order in the prime factorization of a member of A025487 -  is very common in OEIS comments, going back to the 1990s or before.) 

Also, the use of 0 to signify { } in both A212171 and A212172 implies that the second signature of n is also the prime signature of the largest powerful divisor of n, which seems very natural.  But the 0s in A118914 and A212171 aren't the ones I care greatly about, however much sense they seem to make.  

I've worked hard to clarify my own thinking about these issues recently. Thanks for any feedback (especially from Neil). 

Matt Vandermast

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