Math::NumSeq::LemoineCount -- number of representations as P+2*Q for primes P,Q
my $seq = Math::NumSeq::LemoineCount->new;
my ($i, $value) = $seq->next;
This is a count of how many ways i can be represented as P+2*Q for primes P,Q, starting from i=1.
0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 2, 1, 2, 0, 2, 1, 4, 0, ...
For example i=6 can only be written 2+2*2 so just 1 way. But i=9 is 3+2*3=9 and 5+2*2=9 so 2 ways.
Option on_values => 'odd' gives the count on just the odd numbers, starting i=0 for number of ways "1" can be expressed (none),
on_values => 'odd'
0, 0, 0, 1, 2, 2, 2, 2, 4, 2, 3, 3, 3, 4, 4, 2, 5, 3, 4, ...
Lemoine conjectured circa 1894 that all odd i >= 7 can be represented as P+2*Q, which would be a count here always >=1.
Even numbers i are not particularly interesting. An even number must have P even, ie. P=2, so i=2+2*Q for count
count(even i) = 1 if i/2-1 is prime
= 0 if not
See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence classes.
$seq = Math::NumSeq::LemoineCount->new ()
$seq = Math::NumSeq::LemoineCount->new (on_values => 'odd')
Create and return a new sequence object.
$value = $seq->ith($i)
Return the sequence value at $i, being the number of ways $i can be represented as P+2*Q for primes P,Q. or with the on_values=>'odd' option the number of ways for 2*$i+1.
This requires checking all primes up to $i or 2*$i+1 and the current code has a hard limit of 2**24 in the interests of not going into a near-infinite loop.
Math::NumSeq, Math::NumSeq::Primes, Math::NumSeq::GoldbachCount
Copyright 2012, 2013, 2014, 2016, 2019, 2020 Kevin Ryde
Math-NumSeq is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version.
Math-NumSeq is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.
You should have received a copy of the GNU General Public License along with Math-NumSeq. If not, see <http://www.gnu.org/licenses/>.
To install Math::NumSeq, copy and paste the appropriate command in to your terminal.
perl -MCPAN -e shell
For more information on module installation, please visit the detailed CPAN module installation guide.