1Math::NumSeq::ProthNumbUesresr(3C)ontributed Perl DocumeMnattaht:i:oNnumSeq::ProthNumbers(3)
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6 Math::NumSeq::ProthNumbers -- Proth number sequence
7
9 use Math::NumSeq::ProthNumbers;
10 my $seq = Math::NumSeq::ProthNumbers->new;
11 my ($i, $value) = $seq->next;
12
14 The Proth numbers
15 3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, 81, 97, 113, ...
17 starting i=1
18 being integers of the form k*2^n+1 for some k and n and where k < 2^n.
20 The k < 2^n condition means the values in binary have low half 00..01
22 and high half some value k,
23 binary 1xxx0000000...0001
25 value binary k n 2^n
27 3 11 1 1 2
28 5 101 1 2 4
29 9 1001 2 2 4
30 13 1101 3 2 4
31 17 10001 2 3 8
32 25 11001 3 3 8
33 33 100001 4 3 8
34 41 101001 5 3 8
35 ^^
36 ||
37 k part --++-- low part
38 Taking all k < 2^n duplicates some values, as for example 33 is k=4 n=3
40 as 4*2^3+1 and also k=2 n=4 as 2*2^4+1. This happens for any even k.
41 Incrementing n on reaching k=2^n-1 makes a regular pattern, per "Ith"
42 below.
43
45 See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence
46 classes.
47 "$seq = Math::NumSeq::ProthNumbers->new()"
49 Create and return a new sequence object.
50 Random Access
52 "$value = $seq->ith($i)"
53 Return the $i'th Proth number. The first number is 3 at "$i==1".
54 "$bool = $seq->pred($value)"
56 Return true if $value is a Proth number, meaning is equal to
57 k*2^n+1 for some k and n with k < 2^n.
58 "$i = $seq->value_to_i_estimate($value)"
60 Return an estimate of the i corresponding to $value. This is
61 roughly sqrt(2*$value).
62
64 Next
65 Successive values can be calculated by keeping track of the 2^n power
66 and incrementing k by adding such a power,
67 initial
69 value = 1 # to give 3 on first next() call
70 inc = 2
71 limit = 4
72 next()
74 value += inc
75 if value >= limit
76 inc *= 2 # ready for next time
77 limit *= 4
78 return value
79 Ith
81 Taking the values by their length in bits, the values are
82 11 1 value i=1
84 101 1 value i=2
85 1x01 2 values i=3,4
86 1x001 2 values i=5,6
87 1xx001 4 values i=7 to 10
88 1xx0001 4 values
89 1xxx0001 8 values
90 1xxx00001 8 values
91 For a given 1xxx high part the low zeros, which is the 2^n factor, is
93 the same length and then repeated 1 bigger. That doubling can be
94 controlled by a high bit of the i, so in the following Z is either a
95 zero bit or omitted,
96 1Z1
98 1xZ01
99 1xxZ001
100 1xxxZ0001
101 The ith Proth number can be formed from the bits of the index
103 i+1 = 1zxx..xx binary
105 k = 1xx..xx
106 n = z + 1 + number of x's
107 The first 1zxxx desired is 10, which is had from i+1 starting from i=1.
109 The z second highest bit makes n bigger, giving the Z above present or
110 omitted.
111 For example i=9 is bits i+1=1010 binary which as 1zxx is k=0b110=6,
113 n=0+1+2=3, for value 6*2^3+1=49, or binary 110001.
114 It can be convenient to take the two highest bits of the index i
116 together, so hhxx..xx so hh=2 or 3, then n = hh-1 + number of x's.
117
119 Math::NumSeq, Math::NumSeq::CullenNumbers, Math::NumSeq::WoodallNumbers
120
122 <http://user42.tuxfamily.org/math-numseq/index.html>
123
125 Copyright 2010, 2011, 2012, 2013, 2014, 2016, 2019, 2020 Kevin Ryde
126 Math-NumSeq is free software; you can redistribute it and/or modify it
128 under the terms of the GNU General Public License as published by the
129 Free Software Foundation; either version 3, or (at your option) any
130 later version.
131 Math-NumSeq is distributed in the hope that it will be useful, but
133 WITHOUT ANY WARRANTY; without even the implied warranty of
134 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
135 General Public License for more details.
136 You should have received a copy of the GNU General Public License along
138 with Math-NumSeq. If not, see <http://www.gnu.org/licenses/>.
139perl v5.36.0 2022-07-22 Math::NumSeq::ProthNumbers(3)