[seqfan] Re: Wanted: A085080(72).

[Trizen]

Found only two upper-bounds for n = {384, 768}, which are in the temporary
list of integers <= 1000 for which a(n) is unknown:

   a(384) <= 1281916327741
   a(768) <= 1367088016014857

Daniel

On Sat, Mar 16, 2019 at 3:18 PM <hv at crypt.org> wrote:

> michel.marcus at free.fr wrote:
> :Dear SeqFans,
> :Can anyone extend sequence A085080 ?
> :We are looking for a(n) with n= 72, 200, 288, and some more. See unknown
> terms at A085080.
>
> For n = 72, given k = a^3 b^2, n+k = c^3 d^2, it appears that none of
> a, b, c, d is 2 or 3, and we need a == c (mod 24). What other constraints
> do we have?
>
> The simple program below just generates all combinations of primes in
> 5 .. 100,000 and finds no matches, so I think k > 1.25e12. That takes
> rather a lot of memory though, it'll be hard to push much higher than
> 1.25e14 that way.
>
> Hugo
> ---
> #!/usr/bin/perl -w
> use strict;
> use Math::GMP;
> use Math::Prime::Util qw{ primes };
>
> my $primes = primes(5, 100000);
> my %cubes_24;
> for (@$primes) {
>     my $mod = $_ % 24;
>     push @{ $cubes_24{$mod} }, Math::GMP->new($_) * $_ * $_;
> }
>
> for my $mod (0 .. 23) {
>     my $cubes = $cubes_24{$mod} or next;
>     warn sprintf "Test %s primes mod %s\n", 0 + @$cubes, $mod;
>     my %seen;
>     for my $cube (@$cubes) {
>         for my $p_j (@$primes) {
>             my $k = $cube * $p_j * $p_j;
>                         printf "Match %s and %s\n", $k, $k - 72
>                                         if $seen{$k - 72};
>                         printf "Match %s and %s\n", $k, $k + 72
>                                         if $seen{$k + 72};
>             $seen{$k} = 1;
>         }
>     }
> }
>
>
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