Math::NumSeq::DedekindPsiCumulative -- cumulative Psi function
use Math::NumSeq::DedekindPsiCumulative; my $seq = Math::NumSeq::DedekindPsiCumulative->new; my ($i, $value) = $seq->next;
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.
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)
Math::NumSeq, Math::NumSeq::DedekindPsiSteps, Math::NumSeq::TotientCumulative
http://user42.tuxfamily.org/math-numseq/index.html
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.
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