[seqfan] Wanted: Reader of French, regarding A002497

Matthew Vandermast ghodges14 at comcast.net
Mon Jun 4 22:18:47 CEST 2012


Hello Seqfans,

This might be a good time to ask about sequence A002497 (https://oeis.org/A002497), a sequence with the keywords "nice, easy, more" that appeared in the HIS (like the currently-discussed A002219-A002222).  
It's also been discussed fairly recently on the sequence page.   

The sequence page has a link to a paper by J.-L. Nicolas. I don't read French, but the sequence appears to be constructible as described below, *if* the apparent pattern on p. 157 continues.  (That's p. 25 of the PDF.) 

Consider the multiset in which a real number appears m times iff it is a solution to the formulas 
i.) p / log p 
and/or 
ii.) (p^(k+1) - p^k) / log p (for k>0)

in a total of m ways, where p represents a prime. (Unless I'm seriously mistaken, 2/log 2 = (4 - 2)/log 2 = 2.885... is the only number that appears twice, and no number appears more than twice.) 

1. Let f(n) be the n-th-smallest distinct real number that appears in the multiset. 
2. Let q(n) be the specific prime p in the "log p" that is associated with f(n). I avoid "p(n)" here because that usually means something else. 
3. Let m(n) be the number of times f(n) appears in the multiset (i.e., the multiplicity of f(n) in the multiset).
4. Let g(n) = q(n)^m(n).  This is a prime except for g(2) = 2^2 = 4. 

A002497(n) appears to be the product of the first n terms of g(n).   If this is correct, I think there's room for 3 more terms (thank you if you're reading, Dario Alpern):

A002497(16) = A002497(15) * 43 = 156993135980040360 
A002497(17) = A002497(16) * 2   = 313986271960080720 
A002497(18) = A002497(17) * 47 = 14757354782123793840   

But this is all essentially a guess, of course.  By the way, g(n) is not in the database. (I can't see that q(n) and m(n) would be worth entering.) Thanks for any feedback, and please pardon any notational infelicities.

Regards,
Matt Vandermast


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