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


    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.

Values Type

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

Random Access

$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



Copyright 2011, 2012, 2013, 2014, 2016, 2019, 2020 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 <>.