Question about a function: floor(x/ln(x))

[Andrew Plewe]

On Dec 3, 2007 5:34 PM, Andrew Plewe <aplewe at sbcglobal.net> wrote:
> http://www.research.att.com/~njas/sequences/?q=id%3aA066023%7cid%3aA000578&p
> =1&n=10&fmt=0
> (aside from the first couple of terms, will these sequences ever differ?)

I strongly suspect they will.  It's trivial to prove that a(n) <= n^3
-- it's just that n^21 + 1 is divisible by n^7 + 1 -- but proving that
no smaller number of the form k^7 + 1 will be divisible by n^7 + 1
seems difficult, and in fact seems probably false, but checking by
hand up to a few tens of thousands leaves me without a counterexample.
 Does anyone have a big list somewhere like "prime factorizations of
numbers of the form n^7 + 1"?  I suppose making one up to n = 10^6 or
so shouldn't be too hard.  Would I find a counterexample by brute
force there?  Or is there something cleverer involving properties of n
and how they relate to the prime factorization of n^7 + 1?  Hm, n^7 =
-1 (mod p), so n^14 = 1 (mod p), so I suppose I should be looking at
values of p-1 that are multiples of 14, and therefore values of n that
avoid ... argh, I don't know enough number theory to pursue this
argument any farther.

Anyone have some advice for me?

> http://www.research.att.com/~njas/sequences/?q=id:A061557|id:A000782&fmt=0
> (a couple of terms differ at the beginning)

Yup, they are the same, just shifted by one term.

> http://www.research.att.com/~njas/sequences/?q=id:A129367|id:A000912&fmt=0
> (a couple of terms differ at the beginning)

Again the same except for a shift of one term, so probably  A129367
should be rifo (if I have the lingo right) A000912.
The cross-ref from A129366 will need to be edited a bit to take care
of the shift of one term.

> http://www.research.att.com/~njas/sequences/?q=id:A093668|id:A000926&fmt=0
> (Same values, both conjectured to be finite)
>

I think we've discussed this pair enough already - or at least it's
gotten way beyond my capabilities!  We ended with NJAS asking Eric
Rains to verify that they have been proved the same.

> These two sequences:
>
> http://www.research.att.com/~njas/sequences/?q=id:A004306|id:A000803&fmt=0
>
> are the same from "24" onwards. Will they always be the same after that
> point?
>

Sure looks like it to me.  I don't fully understand the definitions,
but if all the comments/cross-refs/formulas are to be believed, you
can chase down some recurrences for these two that are easy enough to
prove equivalent.

IN particular: A000803 is  a(n+3)=a(n+2)+a(n+1)+a(n)-4.
and A004306 says it is equal to 2 * (A001644(n) + 1), and A001644 is
a(n)=a(n-1)+a(n-2)+a(n-3),
so once we have three consecutive terms the same, x, y, z let's say, then
the next term of A000803 is x+y+z-4,
while the three corresponding terms in A001644 are x/2-1, y/2-1, z/2-1,
so the next term in A001644 is x/2 + y/2 + z/2 - 3,
and so 2*(that + 1) is x + y + z - 4,
so it's all good from then out.

--Joshua Zucker