Kevin Ryde > Math-NumSeq-63 > Math::NumSeq::DedekindPsiCumulative

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Module Version: 63   Source   Latest Release: Math-NumSeq-69

# 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)`

# HOME PAGE

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

# LICENSE

Copyright 2012, 2013 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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