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Talk:Decomposition into weight * level + jump

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Your comments about the decomposition are welcome!

Subclassification of numbers or subclassification of terms of a sequence?

I expressed the mathematical formulae in LaTex, I have been very careful not to introduce errors in them while doing so. I am still trying to figure out the ideas you are expressing, I'll look at the links you provided, although it would be nice if you could add more explanatory text on the page itself. — Daniel Forgues 04:31, 6 January 2011 (UTC)
Formulae checked, no error; "more explanatory text": I will try but I am limited by my English — Rémi Eismann 11:33, 7 January 2011 (UTC)
I do not understand the ideas you are presenting, the motivations behind them and how to interpret the formulae (I converted them to LaTeX without understanding them, so I had to be extra careful in doing so) and the graphs. I will ask for other OEIS contributors to have a look. — Daniel Forgues 03:28, 8 January 2011 (UTC)
I cannot see the motivation behind your classification based on Euclidean division, I don't see to what interesting properties of numbers they are relating to... (I added to the page all the OEIS sequences I found relating to your classification, and some rewritten notes encapsulated in tables that I may remove, if you tell me to do so.) Some other OEIS contributors may see better through it than I do, and possibly add more examples and explanatory text. — Daniel Forgues 07:01, 9 January 2011 (UTC)
I removed "notes encapsulated in tables" and added =Decomposition criterion=. I will try to respond you and, thank you, for all the work you're done on this page — Rémi Eismann 09:23, 9 January 2011 (UTC)
This seems to me to be rather a classification of terms of a sequence as they relate to the next term and possibly the previous term of said sequence using Euclidean division. I don't yet see that it reveals interesting properties, except maybe in a few cases... — Daniel Forgues

Conjecture 9

Is conjecture 9 equivalent to Legendre's conjecture (If Legendre's conjecture is true, the gap (or the jump) between any two successive primes would be \scriptstyle O(\sqrt p) \,[1])? — Rémi Eismann 10:33, 9 January 2011 (UTC)
Check A166363 Number of primes in (n*(log(n))^2..(n+1)*(log(n+1))^2] semi-open intervals, n >= 1.
where the n-th interval length is ~ (log(n+1/2))^2 + 2*log(n+1/2) ~ (log(n))^2 as n goes to infinity
and where those intervals EXTREMELY RARELY fail to contain a prime, e.g.
a(n) = 0 for n = 1,4977,17512,147127,76082969 (and no others up to 10^8.) Look at A166363/graph.
It relates to Shanks conjecture[2], Cramér-Granville conjecture[3] and Wolf Conjecture[4].
This means that the first difference between 2 successive primes near \scriptstyle n (log(n))^2 \, might be \scriptstyle O((log(n))^2) \,, but it is not proven yet (this is the Cramér-Granville conjecture)!
Much stronger than Legendre's conjecture[5]!
Thank you Daniel, A166363:nice sequence! I know that Cramér-Granville conjecture is much stronger than Legendre's conjecture. I think there is a relation between Cramér-Granville conjecture and the highest level of primes classified by level. But my question about the relation between conjecture 9 and Legendre's conjecture is still opened — Rémi Eismann 10:37, 15 January 2011 (UTC)

Notes

  1. Legendre's conjecture on wikipedia [1]
  2. Weisstein, Eric W., Shanks Conjecture, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/ShanksConjecture.html]
  3. Weisstein, Eric W., Cramér-Granville Conjecture, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/Cram%C3%A9r-GranvilleConjecture.html]
  4. Weisstein, Eric W., Wolf Conjecture, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/WolfConjecture.html]
  5. Weisstein, Eric W., Legendre's Conjecture, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/LegendresConjecture.html]
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