1Math::NumSeq::GoldbachCUosuenrt(C3o)ntributed Perl DocumMeanttha:t:iNounmSeq::GoldbachCount(3)
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NAME

6       Math::NumSeq::GoldbachCount -- number of representations as sum of
7       primes P+Q
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SYNOPSIS

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

15       The number of ways each i can be represented as a sum of two primes
16       P+Q, starting from i=1,
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18           0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 2, 1, 2, 0, ...
19           starting i=1
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21       For example i=4 can be represented only as 2+2 so just 1 way.  Or i=10
22       is 3+7 and 5+5 so 2 ways.
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24   Even Numbers
25       Option "on_values => 'even'" gives the count on just the even numbers,
26       starting i=1 for number of ways "2" can be expressed (none),
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28           0, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, 2, 3, 2, 4, 4, ...
29           starting i=1
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31       Goldbach's famous conjecture is that for an even i >= 4 there's always
32       at least one P+Q=i, which would be a count here always >= 1.
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34   Odd Numbers
35       Odd numbers i are not particularly interesting.  An odd number can only
36       be i=2+Prime, so the count is simply
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38           count(odd i) = 1  if i-2 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       "$seq = Math::NumSeq::GoldbachCount->new ()"
46       "$seq = Math::NumSeq::GoldbachCount->new (on_values => 'even')"
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 $i can be
52           represented as a sum of primes P+Q, or with the "on_values=>'even'"
53           option the number of ways for "2*$i".
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55           This requires checking all primes up to $i (or "2*$i") 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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59       "$bool = $seq->pred($value)"
60           Return true if $value occurs as a count.  All counts 0 upwards
61           occur so this is simply integer "$value >= 0".
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SEE ALSO

64       Math::NumSeq, Math::NumSeq::Primes, Math::NumSeq::LemoineCount,
65       Math::NumSeq::PrimeFactorCount
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HOME PAGE

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

71       Copyright 2012, 2013, 2014, 2016, 2019, 2020 Kevin Ryde
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73       Math-NumSeq is free software; you can redistribute it and/or modify it
74       under the terms of the GNU General Public License as published by the
75       Free Software Foundation; either version 3, or (at your option) any
76       later version.
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78       Math-NumSeq is distributed in the hope that it will be useful, but
79       WITHOUT ANY WARRANTY; without even the implied warranty of
80       MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
81       General Public License for more details.
82
83       You should have received a copy of the GNU General Public License along
84       with Math-NumSeq.  If not, see <http://www.gnu.org/licenses/>.
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88perl v5.36.0                      2022-07-22    Math::NumSeq::GoldbachCount(3)
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