1Math::NumSeq::DedekindPUssieCrumCuolnattriivbeu(t3e)d PeMralthD:o:cNuummeSnetqa:t:iDoendekindPsiCumulative(3)
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NAME

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

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

14       The cumulative Dedekind Psi function,
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16           1, 4, 8, 14, 20, 32, 40, 52, 64, 82, 94, 118, ...
17           starting i=1
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19           value = sum n=1 to n=i of Psi(n)
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21       where the Psi function is
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23           Psi(n) =        product          (p+1) * p^(e-1)
24                    prime factors p^e in n
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26       The p+1 means one copy of each distinct prime in n is changed from p to
27       p+1.  This is similar to Euler's totient function phi(n) (see
28       Math::NumSeq::Totient) but phi(n) is p-1 instead of p+1.
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FUNCTIONS

31       See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence
32       classes.
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34       "$seq = Math::NumSeq::DedekindPsiCumulative->new ()"
35           Create and return a new sequence object.
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37       "$i = $seq->value_to_i_estimate($value)"
38           Return an estimate of the i corresponding to $value.  Currently
39           this is based on the asymptotic
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41               value = 15*n^2/(2*Pi^2) + O(n*log(n))
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43           which neglecting the O(n*log(n)) becomes
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45               i ~= sqrt(value) * pi/sqrt(15/2)
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SEE ALSO

48       Math::NumSeq, Math::NumSeq::DedekindPsiSteps,
49       Math::NumSeq::TotientCumulative
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HOME PAGE

52       <http://user42.tuxfamily.org/math-numseq/index.html>
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LICENSE

55       Copyright 2012, 2013, 2014, 2016, 2019, 2020 Kevin Ryde
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57       Math-NumSeq is free software; you can redistribute it and/or modify it
58       under the terms of the GNU General Public License as published by the
59       Free Software Foundation; either version 3, or (at your option) any
60       later version.
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62       Math-NumSeq is distributed in the hope that it will be useful, but
63       WITHOUT ANY WARRANTY; without even the implied warranty of
64       MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
65       General Public License for more details.
66
67       You should have received a copy of the GNU General Public License along
68       with Math-NumSeq.  If not, see <http://www.gnu.org/licenses/>.
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72perl v5.36.0                      2023-01M-a2t0h::NumSeq::DedekindPsiCumulative(3)
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