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


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


This sequence is the how many repeated applications of the Dedekind psi function are required to reach a number of the form 2^x*3^y.


The psi function is

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

The p+1 means that one copy of each distinct prime in n is changed from p to p+1. That p+1 is even, so although the value has increased the prime factors are all less than p. Repeated applying that reduction eventually reaches just primes 2 and 3 in some quantity.

For example i=25 requires 2 steps,

    psi(25) = (5+1)*5 = 30 = 2*3*5
    psi(30) = (2+1)*(3+1)*(5+1) = 72 = 2*2*2*3*3

If i is already 2s and 3s then it's considered no steps are required and the value is 0. For example at i=12=2*2*3 the value is 0.


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

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

Create and return a new sequence object.

Random Access

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

Return the number of repeated applications of the psi function on $i required to reach just factors 2 and 3.

This calculation requires factorizing $i and in the current code after small factors a hard limit of 2**32 is enforced in the interests of not going into a near-infinite loop. Above that the return is undef.


Math::NumSeq, Math::NumSeq::TotientSteps, Math::NumSeq::DedekindPsiCumulative



Copyright 2012, 2013, 2014, 2016, 2019, 2020 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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