NAME
Math::NumSeq::ProthNumbers  Proth number sequence
SYNOPSIS
use Math::NumSeq::ProthNumbers;
my $seq = Math::NumSeq::ProthNumbers>new;
my ($i, $value) = $seq>next;
DESCRIPTION
The Proth numbers
3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, 81, 97, 113, ...
starting i=1
being integers of the form k*2^n+1 for some k and n and where k < 2^n.
The k < 2^n condition means the values in binary have low half 00..01 and high half some value k,
binary 1xxx0000000...0001
value binary k n 2^n
3 11 1 1 2
5 101 1 2 4
9 1001 2 2 4
13 1101 3 2 4
17 10001 2 3 8
25 11001 3 3 8
33 100001 4 3 8
41 101001 5 3 8
^^

k part ++ low part
Taking all k < 2^n duplicates some values, as for example 33 is k=4 n=3 as 4*2^3+1 and also k=2 n=4 as 2*2^4+1. This happens for any even k. Incrementing n on reaching k=2^n1 makes a regular pattern, per "Ith" below.
FUNCTIONS
See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence classes.
Random Access
$value = $seq>ith($i)

Return the
$i
'th Proth number. The first number is 3 at$i==1
. $bool = $seq>pred($value)

Return true if
$value
is a Proth number, meaning is equal to k*2^n+1 for some k and n with k < 2^n. $i = $seq>value_to_i_estimate($value)

Return an estimate of the i corresponding to
$value
. This is roughly sqrt(2*$value).
FORMULAS
Next
Successive values can be calculated by keeping track of the 2^n power and incrementing k by adding such a power,
initial
value = 1 # to give 3 on first next() call
inc = 2
limit = 4
next()
value += inc
if value >= limit
inc *= 2 # ready for next time
limit *= 4
return value
Ith
Taking the values by their length in bits, the values are
11 1 value i=1
101 1 value i=2
1x01 2 values i=3,4
1x001 2 values i=5,6
1xx001 4 values i=7 to 10
1xx0001 4 values
1xxx0001 8 values
1xxx00001 8 values
For a given 1xxx high part the low zeros, which is the 2^n factor, is the same length and then repeated 1 bigger. That doubling can be controlled by a high bit of the i, so in the following Z is either a zero bit or omitted,
1Z1
1xZ01
1xxZ001
1xxxZ0001
The ith Proth number can be formed from the bits of the index
i+1 = 1zxx..xx binary
k = 1xx..xx
n = z + 1 + number of x's
The first 1zxxx desired is 10, which is had from i+1 starting from i=1. The z second highest bit makes n bigger, giving the Z above present or omitted.
For example i=9 is bits i+1=1010 binary which as 1zxx is k=0b110=6, n=0+1+2=3, for value 6*2^3+1=49, or binary 110001.
It can be convenient to take the two highest bits of the index i together, so hhxx..xx so hh=2 or 3, then n = hh1 + number of x's.
SEE ALSO
Math::NumSeq, Math::NumSeq::CullenNumbers, Math::NumSeq::WoodallNumbers
HOME PAGE
http://user42.tuxfamily.org/mathnumseq/index.html
LICENSE
Copyright 2010, 2011, 2012, 2013, 2014, 2016, 2019, 2020 Kevin Ryde
MathNumSeq 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.
MathNumSeq 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 MathNumSeq. If not, see <http://www.gnu.org/licenses/>.