Kevin Ryde > Math-NumSeq > Math::NumSeq::DedekindPsiCumulative

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Module Version: 71   Source  

NAME ^

Math::NumSeq::DedekindPsiCumulative -- cumulative Psi function

SYNOPSIS ^

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

DESCRIPTION ^

The cumulative Dedekind Psi function,

    1, 4, 8, 14, 20, 32, 40, 52, 64, 82, 94, 118, ...
    starting i=1

    value = sum n=1 to n=i of Psi(n)

where the Psi function is

    Psi(n) =        product          (p+1) * p^(e-1)
             prime factors p^e in n

The p+1 means one copy of each distinct prime in n is changed from p to p+1. This is similar to Euler's totient function phi(n) (see Math::NumSeq::Totient) but phi(n) is p-1 instead of p+1.

FUNCTIONS ^

See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence classes.

$seq = Math::NumSeq::DedekindPsiCumulative->new ()

Create and return a new sequence object.

$i = $seq->value_to_i_estimate($value)

Return an estimate of the i corresponding to $value. Currently this is based on the asymptotic

    value = 15*n^2/(2*Pi^2) + O(n*log(n))

which neglecting the O(n*log(n)) becomes

    i ~= sqrt(value) * pi/sqrt(15/2)

SEE ALSO ^

Math::NumSeq, Math::NumSeq::DedekindPsiSteps, Math::NumSeq::TotientCumulative

HOME PAGE ^

http://user42.tuxfamily.org/math-numseq/index.html

LICENSE ^

Copyright 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/>.

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