++ed by:
Kevin Ryde
and 1 contributors

NAME

Math::NumSeq::LiouvilleFunction -- Liouville function sequence

SYNOPSIS

use Math::NumSeq::LiouvilleFunction;
my \$seq = Math::NumSeq::LiouvilleFunction->new;
my (\$i, \$value) = \$seq->next;

DESCRIPTION

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.

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, ...

FUNCTIONS

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 requires factorizing \$i and the current code tries small primes then has a hard limit of 2**32 on \$i, 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.