Math::N...Numbers(3)

📦
version 75 · 2 versions
, in fedora 43

🦋 Math::N...Numbers(3) 🦋

okay so real talk: this page lives in the User Contributed Perl Documentation, and the header truncates the title down to Math::N...Numbers(3) because terminals have zero chill 💀. but bestie, it's really about one very specific little sequence class, and it is so cute. let's go. 🫶

💖 NAME 💖

Math::NumSeq::ProthNumbers -- Proth number sequence ✨

🎀 SYNOPSIS 🎀

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

look at that. four lines. immaculate. no notes. 💅

🌸 DESCRIPTION 🌸

The Proth numbers are literally these:

    3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, 81, 97, 113, ...
    starting i=1

and they're the integers of the form k*2^n+1 for some k and n, where k < 2^n. that condition is the whole personality, bestie.

the k < 2^n condition means the values in binary have a low half of 00..01 and a high half of some value k, like this:

    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

see how the k part and the low part just vibe there together? iconic. 🌈

now here's the messy part 🙈 — taking all k < 2^n duplicates some values. for example 33 is k=4 n=3 as 4*2^3+1 and also k=2 n=4 as 2*2^4+1. and this happens for any even k, no cap. incrementing n on reaching k=2^n-1 makes a regular pattern, per the "Ith" section below. iykyk. ✨

💅 FUNCTIONS 💅

check "FUNCTIONS" in Math::NumSeq for the behaviour that's common to all the sequence classes, bestie — that's the shared lore. 📚

  • "$seq = Math::NumSeq::ProthNumbers->new()" — create and return a shiny new sequence object. fresh out the box. 🎁

🍓 Random Access 🍓

  • "$value = $seq->ith($i)" — return the $i'th Proth number. heads up: the first number is 3 at "$i==1". not zero, bestie, one. don't embarrass yourself. 😭
  • "$bool = $seq->pred($value)" — return true if $value is a Proth number, meaning it's equal to k*2^n+1 for some k and n with k < 2^n. it's giving "are you one of us" energy. 🫶
  • "$i = $seq->value_to_i_estimate($value)" — return an estimate of the i corresponding to $value. it's roughly sqrt(2*$value). yes, roughly. it's an estimate, it said what it said. 💅

⭐ FORMULAS ⭐

Next

successive values can be calculated by keeping track of the 2^n power and incrementing k by adding such a power, like so:

    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

lowkey elegant, highkey obsessed. ✨

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, bestie:

       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. ate. 💅

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. slay. 🌟

👑 SEE ALSO 👑

  • Math::NumSeq
  • Math::NumSeq::CullenNumbers
  • Math::NumSeq::WoodallNumbers

the whole friend group. go say hi. 🫶

🏠 HOME PAGE 🏠

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

📜 LICENSE 📜

Copyright 2010, 2011, 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.

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

bestie the WITHOUT ANY WARRANTY part is in ALL CAPS for a reason — that's the legal version of "i'm not responsible for what happens next", so read it fr fr. 😤

and that's the whole page, periodt 💖 this was Math::N...Numbers(3) from perl v5.42.0, dated 2025-07-25. go forth and compute Proth numbers responsibly. mwah. 🎀🦋💅

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