File Coverage

blib/lib/Math/NumSeq/DedekindPsiCumulative.pm

Criterion	Covered	Total	%
statement	58	58	100.0
branch	5	6	83.3
condition			n/a
subroutine	17	17	100.0
pod	3	3	100.0
total	83	84	98.8

# Copyright 2012, 2013, 2014 Kevin Ryde

# This file is part of Math-NumSeq.
#
# 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 .

package Math::NumSeq::DedekindPsiCumulative;
use 5.004;
use strict;
use Math::NumSeq::Primes;

use vars '$VERSION', '@ISA';
$VERSION = 72;

use Math::NumSeq;
@ISA = ('Math::NumSeq');

use Math::NumSeq::PrimeFactorCount;;
*_prime_factors = \&Math::NumSeq::PrimeFactorCount::_prime_factors;

# uncomment this to run the ### lines
#use Smart::Comments;


# use constant name => Math::NumSeq::__('Dedekind Psi Cumulative');
use constant description => Math::NumSeq::__('Cumulative Dedekind Psi function.');
use constant default_i_start => 1;
use constant characteristic_integer => 1;
use constant characteristic_increasing => 1;
use constant values_min => 1;

# A001615 dedekind psi
# A019268 dedekind psi first of n steps
# A019269 number of steps
# A173290 dedekind psi cumulative
#
use constant oeis_anum => 'A173290';

sub rewind {
  my ($self) = @_;
  $self->{'i'} = $self->i_start;
  $self->{'total'} = 0;
}

sub next {
  my ($self) = @_;

  my $i = $self->{'i'}++;
  return ($i,
          $self->{'total'} += _psi($i));
}

sub _psi {
  my ($n) = @_;
  ### _psi(): $n
  my ($good, @primes) = _prime_factors($n);
  return undef unless $good;

  my $prev = 0;
  foreach my $p (@primes) {
    if ($p != $prev) {
      $n /= $p;
      $n *= $p+1;
      $prev = $p;
    }
  }
  ### $n
  return $n;
}

# v1.02 for leading underscore
use constant 1.02 _PI => 4*atan2(1,1); # similar to Math::Complex pi()

# Enrique Pérez Herrero in the OEIS
# value = 15*n^2/(2*Pi^2) + O(n*log(n))
#
# sqrt(value) = sqrt(15/2)/pi * n
# n = sqrt(value) * pi/sqrt(15/2)
# with pi/sqrt(15/2) ~= 39/34 for the benefit of Math::BigInt which can't
# do BigInt*flonum
#
# very close even neglecting n*log(n)
#
sub value_to_i_estimate {
  my ($self, $value) = @_;
  if ($value <= 1) { return 1; }
  return int (sqrt(int($value)) * 39/34);
}

1;
__END__