Math::NumSeq::LiouvilleFunction -- Liouville function sequence
use Math::NumSeq::LiouvilleFunction; my $seq = Math::NumSeq::LiouvilleFunction->new; my ($i, $value) = $seq->next;
The Liouville function parity of the prime factors of i,
1, -1, -1, 1, -1, 1, -1, -1, 1, 1, -1, -1, -1, 1, 1, ... starting i=1
being
1 if i has an even number of prime factors -1 if i has an odd number of prime factors
The sequence starts from i=1 which is taken to be no prime factors, ie. zero, which is even, hence value 1. Then i=2 and i=3 are -1 since they have one prime factor (they're primes), and i=4 is value 1 because it's 2*2 which is an even number of prime factors (two 2s).
This parity is similar to the MobiusFunction
, but here repeated prime factors are included, whereas in MobiusFunction
they give a value 0.
The values_type
parameter can change the two values returned for even or odd prime factors. "0,1" gives 0 for even and 1 for odd, the same as the count mod 2,
values_type => '0,1' 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, ...
Or "1,0" the other way around, 1 for even, 0 for odd,
values_type => '1,0' 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, ...
See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence classes.
$seq = Math::NumSeq::LiouvilleFunction->new ()
$seq = Math::NumSeq::LiouvilleFunction->new (values_type => $str)
Create and return a new sequence object. Optional values_type
(a string) can be
"1,-1" 1=even, -1=odd (the default) "0,1" 0=even, 1=odd "1,0" 1=even, 0=odd
$value = $seq->ith($i)
Return the Liouville function of $i
, being 1 or -1 (or other values_type
) according to the number of prime factors in $i
.
This calculation requires factorizing $i
and in the current code after small factors a hard limit of 2**32 is enforced in the interests of not going into a near-infinite loop.
$bool = $seq->pred($value)
Return true if $value
occurs in the sequence, which simply means 1 or -1, or the two values_type
values.
Math::NumSeq, Math::NumSeq::MobiusFunction, Math::NumSeq::PrimeFactorCount
http://user42.tuxfamily.org/math-numseq/index.html
Copyright 2011, 2012, 2013, 2014 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/>.