1Math::NumSeq::LemoineCoUusnetr(3C)ontributed Perl DocumeMnattaht:i:oNnumSeq::LemoineCount(3)
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NAME

6       Math::NumSeq::LemoineCount -- number of representations as P+2*Q for
7       primes P,Q
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SYNOPSIS

10        use Math::NumSeq::LemoineCount;
11        my $seq = Math::NumSeq::LemoineCount->new;
12        my ($i, $value) = $seq->next;
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DESCRIPTION

15       This is a count of how many ways i can be represented as P+2*Q for
16       primes P,Q, starting from i=1.
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18           0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 2, 1, 2, 0, 2, 1, 4, 0, ...
19           starting i=1
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21       For example i=6 can only be written 2+2*2 so just 1 way.  But i=9 is
22       3+2*3=9 and 5+2*2=9 so 2 ways.
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24   Odd Numbers
25       Option "on_values => 'odd'" gives the count on just the odd numbers,
26       starting i=0 for number of ways "1" can be expressed (none),
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28           0, 0, 0, 1, 2, 2, 2, 2, 4, 2, 3, 3, 3, 4, 4, 2, 5, 3, 4, ...
29           starting i=0
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31       Lemoine conjectured circa 1894 that all odd i >= 7 can be represented
32       as P+2*Q, which would be a count here always >=1.
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34   Even Numbers
35       Even numbers i are not particularly interesting.  An even number must
36       have P even, ie. P=2, so i=2+2*Q for count
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38           count(even i) = 1 if i/2-1 is prime
39                         = 0 if not
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FUNCTIONS

42       See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence
43       classes.
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45       "$seq = Math::NumSeq::LemoineCount->new ()"
46       "$seq = Math::NumSeq::LemoineCount->new (on_values => 'odd')"
47           Create and return a new sequence object.
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49   Random Access
50       "$value = $seq->ith($i)"
51           Return the sequence value at $i, being the number of ways $i can be
52           represented as P+2*Q for primes P,Q. or with the "on_values=>'odd'"
53           option the number of ways for "2*$i+1".
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55           This requires checking all primes up to $i or "2*$i+1" and the
56           current code has a hard limit of 2**24 in the interests of not
57           going into a near-infinite loop.
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SEE ALSO

60       Math::NumSeq, Math::NumSeq::Primes, Math::NumSeq::GoldbachCount
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HOME PAGE

63       <http://user42.tuxfamily.org/math-numseq/index.html>
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LICENSE

66       Copyright 2012, 2013, 2014, 2016, 2019, 2020 Kevin Ryde
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68       Math-NumSeq is free software; you can redistribute it and/or modify it
69       under the terms of the GNU General Public License as published by the
70       Free Software Foundation; either version 3, or (at your option) any
71       later version.
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73       Math-NumSeq is distributed in the hope that it will be useful, but
74       WITHOUT ANY WARRANTY; without even the implied warranty of
75       MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
76       General Public License for more details.
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78       You should have received a copy of the GNU General Public License along
79       with Math-NumSeq.  If not, see <http://www.gnu.org/licenses/>.
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83perl v5.36.0                      2023-01-20     Math::NumSeq::LemoineCount(3)
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