a property of multisets: unit fractions with a twist

[Giovanni Resta]

hv at crypt.org wrote:

> hv at crypt.org wrote:
> :%S A000001 1,1,5,43,875,49506
> 
> Bother, my crude code was cruder than I had realised: both the last two
> terms are wrong. Corrected terms shown in the copy below. I'm not sure
> how I'd go about calculating a(7), I'd welcome thoughts on that.
> 
> %I A000001 
> %S A000001 1,1,5,43,876,49511

Now I realize that the sequence you are pursuing is already in OEIS:

A118085
Number of ways 2 is a product of n superparticular ratios,
without regard to order. A superparticular ratio is a ratio of the
form m/(m-1).
1, 1, 5, 43, 876, 49513


bye,
giovanni




I'd consider that valid, but it's by no means the most "compact:

1, 2, 3, 7, 43, 1892,...

While looking up Andrica's conjecture, I found this note:

5) p   / p  <= 5/3, and the maximum occurs at n = 2.



I checked the reference and the paper was published. I assume, therefore,
that the proof is accepted. Let p_n = 2 and maximize the sequence by setting
p_n+1/p_n = 5/3. Proceed as follows:

a(1) = 2
a(2) = x/2 = 5/3, by cross-multiplying: 3x = 10, then ceiling(10/3) = 4
a(3) = x/4 = 5/3, 3x = 20, ceiling(20/3) = 7
a(4) = x/7 = 5/3, 3x = 35, ceiling(35/3) = 12
a(5) = x/12 = 5/3, 3x = 60 = 20
a(6) = x/20 = 5/3, 3x = 100, ceiling(100/3) = 34
a(7) = x/34 = 5/3, 3x = 170, ceiling(170/3) = 57
a(8) = x/57 = 5/3, 3x = 285 = 95

and so on. This is ultimately, I believe, more "compact" than A052548:

3, 4, 6, 10, 18, 34, 66, 130, 258, 514, 1026...

If I've constructed the sequence correctly, than their is either a prime
between a(n) and a(n+1), or a(n+1) is prime.




-----Original Message-----
maximilian.hasler at gmail.com
between a(n) and a(n+1)

> Thus a clarification; generating the sequence shouldn't
> require that we know which numbers are prime.

That's a bit vague, is it not? Let's define a(1) = 2 and
a(n+1) as the smallest number that can't be written
as the product of numbers in {1,2,3,...,a(n)} plus 1.
That way we should always get a prime in between
without knowing what primes are.

An almost number-theoretic free sequence could be defined
as such: Andrica's conjecture implies p_(n+1) - p_(n) < 2*sqrt(p_n) +1.
Assuming its truth and defining a(1) = 2 and
a(n+1)  = ceiling(a(n) + 2*sqrt(a(n)) + 1) should give such
a sequence. (Are there any nice proven bounds like that?)

Stefan