[seqfan] Re: Testing potential squares

[Hugo Pfoertner]

With respect to A085635 and the related A084848, A085635(1) should
definitely be 1. A085635 just gives the positions of decreasing minima of
the ratio R=A000224(n)/n:
n A000224(n) R
1 1  1.0000 <-
2 2  1.0000
3 2  0.6667 <-
4 2  0.5000 <-
5 3  0.6000
6 4  0.6667
7 4  0.57143
8 3  0.3750 <-
...
Therefore also A084848(1) should be 1 instead of 2.

The PARI programs in A085635 also need a corresponding correction.

And, of course, if you have more terms than those shown for both sequences,
you should submit b-files. I myself have only about 90 terms.

On Wed, Aug 22, 2018 at 12:44 PM Keith F. Lynch <kfl at keithlynch.net> wrote:

> To rule out most of the remaining non-squares, I searched for numbers
> that had a lower proportion of quadratic residues than any smaller
> number (e.g. 1441440, the first number such that fewer than one
> percent of numbers are quadratic residues modulo it), intending to
> create a table of residues mod such a number and test every potential
> square by calculating it mod that number and seeing if it was in the
> table.  After calculating the first few numbers that have a record
> low proportion of quadratic residues, I looked them up in OEIS, but
> couldn't find them.  Later I found they're at A085635; I had missed
> it because it lists the first term as 2 when I think it should be 1.
>
> Should I correct that, or is there a reason it's 2?  Thanks.  I also
> have more terms to add to it and to its companion sequence A084848.
> They currently show 44 terms, the last of which is around a hundred
> thousand and has a ratio of about 50.  I've computed 193 terms so
> far, of which the last is about two thirds of a trillion and has a
> ratio of about 1588.
>