NAME
Math::NumSeq::LemoineCount  number of representations as P+2*Q for primes P,Q
SYNOPSIS
use Math::NumSeq::LemoineCount;
my $seq = Math::NumSeq::LemoineCount>new;
my ($i, $value) = $seq>next;
DESCRIPTION
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, ...
starting i=1
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.
Odd Numbers
Option on_values => 'odd'
gives the count on just the odd numbers, starting i=0 for number of ways "1" can be expressed (none),
0, 0, 0, 1, 2, 2, 2, 2, 4, 2, 3, 3, 3, 4, 4, 2, 5, 3, 4, ...
starting i=0
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
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/21 is prime
= 0 if not
FUNCTIONS
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.
Random Access
$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 theon_values=>'odd'
option the number of ways for2*$i+1
.This requires checking all primes up to
$i
or2*$i+1
and the current code has a hard limit of 2**24 in the interests of not going into a nearinfinite loop.
SEE ALSO
Math::NumSeq, Math::NumSeq::Primes, Math::NumSeq::GoldbachCount
HOME PAGE
http://user42.tuxfamily.org/mathnumseq/index.html
LICENSE
Copyright 2012, 2013, 2014, 2016, 2019, 2020 Kevin Ryde
MathNumSeq 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.
MathNumSeq 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 MathNumSeq. If not, see <http://www.gnu.org/licenses/>.