1Math::NumSeq::MobiusFunUcsteironC(o3n)tributed Perl DocuMmaetnht:a:tNiuomnSeq::MobiusFunction(3)
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NAME

6       Math::NumSeq::MobiusFunction -- Mobius function sequence
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SYNOPSIS

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

14       The sequence of the Mobius function,
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16           1, -1, -1, 0, -1, 1, ...
17           starting i=1
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19       Each value is
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21           1   if i has an even number of distinct prime factors
22           -1  if i has an odd number of distinct prime factors
23           0   if i has a repeated prime factor
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25       The sequence starts from i=1 and it's reckoned as no prime factors, ie.
26       0 factors, which is considered even, hence value=1.  Then i=2 and i=3
27       are value=-1 since they have one prime factor (they're primes), and i=4
28       is value=0 because it's 2*2 which is a repeated prime 2.
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FUNCTIONS

31       See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence
32       classes.
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34       "$seq = Math::NumSeq::MobiusFunction->new ()"
35           Create and return a new sequence object.
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37   Random Access
38       "$value = $seq->ith($i)"
39           Return the Mobius function of $i, being 1, 0 or -1 according to the
40           prime factors of $i.
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42           This calculation requires factorizing $i and in the current code
43           small primes are checked then a hard limit of 2**32 is placed on
44           $i, in the interests of not going into a near-infinite loop.
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46       "$bool = $seq->pred($value)"
47           Return true if $value occurs in the sequence, which means simply 1,
48           0 or -1.
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SEE ALSO

51       Math::NumSeq, Math::NumSeq::LiouvilleFunction,
52       Math::NumSeq::PrimeFactorCount
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54       "moebius" in Math::Prime::Util
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HOME PAGE

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

60       Copyright 2010, 2011, 2012, 2013, 2014, 2016, 2019, 2020 Kevin Ryde
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62       Math-NumSeq is free software; you can redistribute it and/or modify it
63       under the terms of the GNU General Public License as published by the
64       Free Software Foundation; either version 3, or (at your option) any
65       later version.
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67       Math-NumSeq is distributed in the hope that it will be useful, but
68       WITHOUT ANY WARRANTY; without even the implied warranty of
69       MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
70       General Public License for more details.
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72       You should have received a copy of the GNU General Public License along
73       with Math-NumSeq.  If not, see <http://www.gnu.org/licenses/>.
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77perl v5.36.0                      2023-01-20   Math::NumSeq::MobiusFunction(3)
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