LINKS
Pontus von Brömssen, <a href="/A363098/b363098_1.txt">Table of n, a(n) for n = 1..10000</a>
Discussion
Thu May 25
15:14
OEIS Server: Installed first b-file as b363098.txt.
LINKS
Pontus von Brömssen, <a href="/A363098/b363098_1.txt">Table of n, a(n) for n = 1..10000</a>
COMMENTS
Factorization into primitive terms of A363063 is not unique. The first counterexample is 1728 = 768 864 * 2 = 12^3.
Discussion
Sun May 21
10:35
Peter Munn: Do you already have a program, Pontus?'
10:54
David A. Corneth: Would "a(n) is the smallest term of A363063 divisible by prime(n)" be an interesting sequence? Not finding that in OEIS
10:57
David A. Corneth: Seems like a subset of A347284
11:16
Pontus von Brömssen: @Peter: I have a program, but it's basically the same as my program for A363063, just filtering out primitive terms afterwards. The filtering takes much more time than generating the terms of A363063; maybe there is a smarter way to do it? Anyway, I just prepared a b-file with 10000 terms.
11:18
Pontus von Brömssen: Regarding 3-smooth terms, there are just 7 of them among the first 10000 terms: 4 in the data, then 2^27*3^17 (as you noted, Peter), then 2^46*3^29 and 2^65*3^41.
11:25
Pontus von Brömssen: @David: "smallest term divisible by prime(n)" should be the same sequence as "terms with new gpf". There's already a comment (by Peter) that these (and only these) are in A347284. Could certainly be interesting enough to submit this sequence.
11:47
Peter Munn: #11:18 Yes, this was the intent of my "converging from above" variant of A005663/A005664. It would start (1/0,) 2/1, 5/3, 8/5, 27/17, 46/29, 65/41, 149/94, 233/147, ...
12:00
Peter Munn: I'm sure we will come up with a smarter way. I'm a big believer in starting with a simple approach -- gives a data set against which to check the more error-prone smart program.
12:04
Peter Munn: Anyway, I'm happy for this to be proposed.
12:56
Peter Munn: One more thing I ought to say:
Expanding on 10:17, I suspect a fairly comprehensive characterisation of the terms will need some work, and I think the proof of my statement about "infinitely many terms with gpf = p" would be better explained in that context. But in summary, let b denote the infinite sequence of 3-smooth terms, which is calculated using the convergent fractions mentioned in pink box 11:47. For prime p > 3, let k_p be the first term in A347284 (equivalently, here) that has p as gpf. Then {m : m = lcm(b(i), k_p), m >= k_p, i >= 1 } is a suitable infinite set of primitive terms. (It is not difficult to show that every nonunit A363063 factor of every qualifying m must be divisible by b(i) and k_p.)
COMMENTS
Factorization into primitive terms of A363063 is not unique. The first counterexample is 1728 = 768 * 2 = 12^3.
EXAMPLE
4 is in A363063, but is not a term here, because 2 is in A363063 and 2 * 2 = 4.
720 is the first term of A363063 that is divisible by 5, from which we deduce 720 is not a product of nonunit terms of A363063. So 720 is a term here.
Discussion
Sun May 21
10:17
Peter Munn: Characterisation of the terms seems an interesting challenge, so might be better left for later. Also, OEIS seems not yet to have some related sequences that would provide added-value for this, such as the "converging from above" variant of A005663/A005664.
COMMENTS
For every odd prime p there are infinitely many terms whose greatest prime factor is p. The first Reading along the sequence, we see a term with any given a new greatest prime factor if and only if it is also in A347284.
