1Math::NumSeq::DedekindPUssieSrteCposn(t3r)ibuted Perl DoMcautmhe:n:tNautmiSoenq::DedekindPsiSteps(3)
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NAME

6       Math::NumSeq::DedekindPsiSteps -- psi function until 2^x*3^y
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SYNOPSIS

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

14       This sequence is the how many repeated applications of the Dedekind psi
15       function are required to reach a number of the form 2^x*3^y.
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17           0,0,0,0,1,0,1,0,0,1,1,0,2,1,1,0,1,0,2,1,1,1,1,0,2,2,0,1,2,...
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19       The psi function is
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21           psi(n) =        product          (p+1) * p^(e-1)
22                    prime factors p^e in n
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24       The p+1 means that one copy of each distinct prime in n is changed from
25       p to p+1.  That p+1 is even, so although the value has increased the
26       prime factors are all less than p.  Repeated applying that reduction
27       eventually reaches just primes 2 and 3 in some quantity.
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29       For example i=25 requires 2 steps,
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31           psi(25) = (5+1)*5 = 30 = 2*3*5
32           then
33           psi(30) = (2+1)*(3+1)*(5+1) = 72 = 2*2*2*3*3
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35       If i is already 2s and 3s then it's considered no steps are required
36       and the value is 0.  For example at i=12=2*2*3 the value is 0.
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FUNCTIONS

39       See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence
40       classes.
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42       "$seq = Math::NumSeq::DedekindPsiSteps->new ()"
43           Create and return a new sequence object.
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45   Random Access
46       "$value = $seq->ith($i)"
47           Return the number of repeated applications of the psi function on
48           $i required to reach just factors 2 and 3.
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50           This calculation requires factorizing $i and in the current code
51           after small factors a hard limit of 2**32 is enforced in the
52           interests of not going into a near-infinite loop.  Above that the
53           return is "undef".
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SEE ALSO

56       Math::NumSeq, Math::NumSeq::TotientSteps,
57       Math::NumSeq::DedekindPsiCumulative
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HOME PAGE

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

63       Copyright 2012, 2013, 2014, 2016, 2019 Kevin Ryde
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65       Math-NumSeq is free software; you can redistribute it and/or modify it
66       under the terms of the GNU General Public License as published by the
67       Free Software Foundation; either version 3, or (at your option) any
68       later version.
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70       Math-NumSeq is distributed in the hope that it will be useful, but
71       WITHOUT ANY WARRANTY; without even the implied warranty of
72       MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
73       General Public License for more details.
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75       You should have received a copy of the GNU General Public License along
76       with Math-NumSeq.  If not, see <http://www.gnu.org/licenses/>.
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80perl v5.34.0                      2021-07-22 Math::NumSeq::DedekindPsiSteps(3)
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