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Math-NumSeq-21
River stage zero No dependents
3 ++
/ Math::NumSeq::ProthNumbers

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, etc, being integers k*2^n+1 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
      3      11   
      5      101    
      9     1001
     13     1101
     17     10001
     25     11001
     33    100001
             ^^
             ||
    k part --++-- low part

FUNCTIONS

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

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

Create and return a new sequence object.

$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.

FORMULAS

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 = hh-1 + number of x's.

SEE ALSO

Math::NumSeq, Math::NumSeq::CullenNumbers, Math::NumSeq::WoodallNumbers

HOME PAGE

http://user42.tuxfamily.org/math-numseq/index.html

LICENSE

Copyright 2010, 2011 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.

You should have received a copy of the GNU General Public License along with Math-NumSeq. If not, see <http://www.gnu.org/licenses/>.