Discussion
Thu Apr 27
07:10
Ralf Steiner: In my opinion there is a convention for the multiplication. Thus I don't think that a * is necessary there.
07:16
Ralf Steiner: well-ordered means as usually ..., 5,7,11,13, ...
07:29
Antti Karttunen: Ralf: Just edited https://oeis.org/draft/A285388
Is it guaranteed that all of its terms are odd? (then my new "formula" (based on yours) should be valid, otherwise not).
07:31
Antti Karttunen: So "the upper part of at least n in i well-ordered prime factors p_i(n)" means at least n greatest prime factors (with or without multiplicity?) of n ?
07:40
Ralf Steiner: Antti: A285388 is approved.
07:50
Ralf Steiner: Antti: To your question. By my proof sketch YES. But Mathematica gives: OddQ[Numerator[2^(1 - 2 n^2) n Binomial[2 n^2, n^2]]]
=> FALSE
==>> Note: Expressions that represent odd integers but do not evaluate explicitly will still give False.
08:17
Ralf Steiner: Antti: The at least n greatest prime factors are without multiplicity - these are single each.
16:19
David A. Corneth: I look forward to seeing the paper Ralf! If you need help, let me know.
16:28
Ralf Steiner: Thanks!
16:33
Ralf Steiner: David: I am still looking for the original paper of Legendre's Conjecture.
17:08
David A. Corneth: Okay, I'll help looking
17:51
David A. Corneth: Not found yet. In his list of publications there is one article I think might have it; "Essai sur la Théorie des Nombres", written by Legendre and published by Duprat in 1798. I can't find it online but it is said to be in a library near me.
17:57
David A. Corneth: Here's a scanned version: http://www.biodiversitylibrary.org/item/58163#page/6/mode/1up
Fri Apr 28
00:43
Ralf Steiner: Great - thank you!
01:32
Antti Karttunen: Dear Dr. Steiner, Could I replace your sentence in the beginning of https://oeis.org/draft/A285388 "a(n) has at least n prime factors, each A285738(n) less than 2*n^2" WITH THIS "a(n) has at least n prime factors, the largest of them which is A285738(n), which is the greatest prime less than 2*n^2" ? Or is there more contents into it?
01:34
Antti Karttunen: Please see also my new formula in https://oeis.org/draft/A285406
Fundamentally, it is based on Vladimir Shevelev's Jul 20 2009 formula in A000984, that A007814(A000984(n)) = A000120(n).
02:10
Ralf Steiner: Antti: replaced
02:31
Ralf Steiner: David: The book has 562 pages. Do you know where the essential sentences for his conjecture are written?
02:34
Antti Karttunen: @02:10. Ralf: Thanks!
02:41
Ralf Steiner: I don't find (using search function) a term like (n+1) in his book.
02:45
Ralf Steiner: Maybe his conjecture is "between the lines", but where?
02:59
Ralf Steiner: Antti, David: Please visit my latest note at A285786 with a question regarding the original conjecture.
06:21
Ralf Steiner: After studying Legendre's book from 1797 a bit I am sure that the interval (2(n-1)^2, n^2 ) used in my proof also fullfil his conjecture.
07:15
Ralf Steiner: I propose the sequence is ready now to become the status "approved".
