# [seqfan] Re: Second signature: Two language questions

Charles Greathouse charles.greathouse at case.edu
Mon Jun 11 18:45:23 CEST 2012

```> a. Prime, secondary, tertiary...k-ary signature
> b.  Prime, second, third...k-th signature

I'm generally uneasy with using this kind of novel term to name a
sequence.  We can't expect that someone will know what a secondary (or
second) signature is -- it's enough to ask people to know what a prime
signature is!  I much prefer definitions which can be understood
without specialized vocabulary when possible.

As to the particular choice of phrasing, without respect to naming
sequences, I don't know that it conveys the right sense.  It's not
that the "secondary signature" looks at "secondary" objects like the
prime signature looks at primes, but that it excludes exponents < 2.
(If it looked at, say, prime squares then it couldn't distinguish 9
from 27 -- 9 appears only to the first power, right?)  So perhaps it
should be along the lines of "2-truncated prime signature".

> 2. This is a more general OEIS usage question.  How OK is it to call the smallest or least integers of a given property the *first* integers with that property?

Probably fine.  I'm not sure that I see the distinction -- they're
first with respect to <, not the first by date or the first in
alphabetical order.  I think the choice between the two is mostly
stylistic.  I tend to use "Least n such that ...", personally.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University

On Mon, Jun 11, 2012 at 12:18 PM, Matthew Vandermast
<ghodges14 at comcast.net> wrote:
> Before I start submitting these sequences...
>
> 1. I meant to ask this earlier, but: What do people (especially Neil) think is the better series of names for the ordered list (and/or multiset) of the exponents  >k in the prime factorization of n?
>
> a. Prime, secondary, tertiary...k-ary signature
> (follows "primus, secundus, tertius...")
>
> b.  Prime, second, third...k-th signature
> (might be an acceptable usage that also won't make anyone too queasy, and saves syllables and keystrokes)
>
> Among higher (or "post-prime," if you will) signatures, the second/secondary signature seems to hold at least 90% of the mathematical interest, and probably more than that.  I'm not personally interested in submitting sequences of third/tertiary signatures or anything higher.
>
> 2. This is a more general OEIS usage question.  How OK is it to call the smallest or least integers of a given property the *first* integers with that property?
>
> From sequences titled, e.g., "Sum of first n primes" or "Sum of first n cubes," I gather this is at least borderline OK, which is more than fine with me. There is a natural assumption (also implicit in the phrase "natural density") that the "natural" way to travel the number line is in the positive direction. But this is obviously subject to correction.
>
> Take A181800, for example.   These integers could be referenced as the "least" or "smallest" integers of each second(ary) signature.  But I don't think the interesting thing about them is their petite size compared to other members of their second(ary) signatures. I think the interesting thing is that they're the first.  (This could be spelled out by saying they're the first members of A000027 of each second(ary) signature, but if this is not necessary, it is probably not desirable.)
>
> Maybe Neil's answer will be roughly equivalent to: "Yes, that's borderline OK, but two or three equally significant departures from strict technical rectitude in the same sequence would be pushing it."  (I'm not sure whether what I'm asking about *is* a departure from technical rectitude.) In any case, I hope it's a question of general interest.
>
> Thanks,
> Matt Vandermast
>
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