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

[hv at crypt.org]

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;
        }
    }
}