[seqfan] Second signature: Two language questions

Matthew Vandermast ghodges14 at comcast.net
Mon Jun 11 18:18:45 CEST 2012

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.

Matt Vandermast

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