1Math::NumSeq::RepdigitRUasdeirx(C3o)ntributed Perl DocumMeanttha:t:iNounmSeq::RepdigitRadix(3)
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NAME

6       Math::NumSeq::RepdigitRadix -- radix in which i is a repdigit
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SYNOPSIS

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

14       The radix in which i is a repdigit,
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16           2, 0, 0, 2, 3, 4, 5, 2, 3, 8, 4, 10, etc
17           starting i=0
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19       i=0 is taken to be a repdigit "00" in base 2.  i=1 and i=2 are not
20       repdigits in any radix.  Then i=3 is repdigit "11" in base 2.  Any i>=3
21       is at worst a repdigit "11" in base i-1, but may be a repdigit in a
22       smaller base.  For example i=8 is "22" in base 3.
23
24       Is this behaviour for i=0,1,2 any good?  Perhaps it will change.
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FUNCTIONS

27       See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence
28       classes.
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30       "$seq = Math::NumSeq::RepdigitRadix->new ()"
31           Create and return a new sequence object.
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33   Random Access
34       "$value = $seq->ith($i)"
35           Return the radix in which $i is a repdigit.
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37           The current code relies on factorizing $i and a hard limit of 2**32
38           is placed on $i in the interests of not going into a near-infinite
39           loop.
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FORMULAS

42   ith() Value
43       "ith()" looks for the smallest radix r for which there's a digit d and
44       length len satisfying
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46           i = d * repunit(len)
47           i = d * (r^(len-1) + r^(len-2) + ... + r^2 + r + 1)
48
49       The current approach is to consider repdigit lengths successively from
50       log2(i) downwards and candidate digits d from among the divisors of i.
51
52           for len=log2(i) down to 2
53             for d each divisor of i, descending
54               r = nthroot(i/d, len-1)
55               if r >= r_found then next len
56               if r <= d then next divisor
57               if (r^len-1)/(r-1) == i/d then r_found=r, next len
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59           if no r_found then r_found = i-1
60
61       For a given d the radix r to give i would be
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63           i/d = r^(len-1) + ... + r + 1
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65       but it's enough to calculate
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67           i/d = r^(len-1)
68           r = floor nthroot(i/d, len-1)
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70       and then power up to see if it gives the desired i/d.
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72           repunit(len) = r^(len-1) + ... + r + 1
73                        = (r^len - 1) / (r-1)
74           check if equals i/d
75
76       floor(nthroot()) is never too small, since an r+1 from it would give
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78           (r+1)^(len-1) = r^(len-1) + binomial*r^(len-2) + ... + 1
79                         > r^(len-1) +          r^(len-2) + ... + 1
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81       Divisors are taken in descending order so the radices r are in
82       increasing order.  So if a repdigit is found in a given len then it's
83       the smallest of that length and can go on to other lengths.
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85       The lengths can be considered in any order but the current code goes
86       from high to low since a bigger length means a smaller maximum radix
87       within that length (occurring when d=1, ie. a repunit), so it might
88       establish a smaller "r_found" and a smaller r_found limits the number
89       of divisors to be tried in subsequent lengths.  But does that actually
90       happen often enough to make any difference?
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92   ith() Other Possibilities
93       When len is even the repunit part r^(len-1)+...+1 is a multiple of r+1.
94       Can that cut the search?  For a given divisor the r is found easily
95       enough by nthroot, but maybe i with only two prime factors can never be
96       an even length>=4 repdigit, or something like that.
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SEE ALSO

99       Math::NumSeq, Math::NumSeq::RepdigitAny
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HOME PAGE

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

105       Copyright 2010, 2011, 2012, 2013, 2014, 2016, 2019 Kevin Ryde
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107       Math-NumSeq is free software; you can redistribute it and/or modify it
108       under the terms of the GNU General Public License as published by the
109       Free Software Foundation; either version 3, or (at your option) any
110       later version.
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112       Math-NumSeq is distributed in the hope that it will be useful, but
113       WITHOUT ANY WARRANTY; without even the implied warranty of
114       MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
115       General Public License for more details.
116
117       You should have received a copy of the GNU General Public License along
118       with Math-NumSeq.  If not, see <http://www.gnu.org/licenses/>.
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122perl v5.34.0                      2021-07-22    Math::NumSeq::RepdigitRadix(3)
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