Bounding the density of abundant numbers and covering numbers.

This repository contains high-performance C++ programs for computing rigorous density bounds of two specific classes of integers.

Programs

Abundant Number Density Calculator (AbunDens)

Computes rigorous upper and lower bounds for the density of abundant numbers. A positive integer n is abundant if σ(n)/n ≥ 2, where σ(n) is the sum of divisors function.

Covering Number Density Calculator (CovDens)

Computes rigorous upper bounds for the density of covering numbers. A positive integer n is covering if a distinct covering system of the integers can be formed using the divisors of n greater than one as moduli.

Features

• Rigorous Interval Arithmetic: Uses high-precision MPFR arithmetic with configurable precision for calculation of moments
• Rigorous Interval Arithmetic: Uses long doubles with consistent rounding modes set for density calculations
• Parallel Processing: OpenMP-based parallelization for efficient computation on multi-core systems
• Persistent Computation: Automatic save/resume functionality for long-running calculations

Prerequisites

• C++17 compatible compiler (GCC 7+ recommended)
• GMP (GNU Multiple Precision Arithmetic Library)
• MPFR (Multiple Precision Floating-Point Reliably)
• OpenMP for parallel computation

Installing Dependencies

Ubuntu/Debian:

sudo apt-get install build-essential libgmp-dev libmpfr-dev

macOS (via Homebrew):

brew install gmp mpfr

CentOS/RHEL:

sudo yum install gcc-c++ gmp-devel mpfr-devel

Building

The project uses a comprehensive Makefile with multiple build targets:

# Build both programs (optimized)
make

# Build individual programs
make abundant
make covering

## Usage

Both programs accept similar command-line arguments:

```bash
# Basic usage with default precision (80 bits)
./bin/abundens <zpow> <numPrimes>

# Custom precision
./bin/abundens <zpow> <numPrimes> [precision]

# Examples
./bin/abundens 15 100000        # Default 80-bit precision
./bin/abundens 15 100000 120    # 120-bit precision

Parameters

• zpow: Search depth parameter

  • Controls search depth as the parameter z=2^zpow
  • Higher values = more precision but slower computation

• numPrimes: Number of consecutive primes to consider

  • More primes = better bounds but longer computation time

• precision: MPFR precision in bits (optional, default: 80, range: 53-1000)

  • 80 bits seems sufficent for current calculations.

Mathematical Background

Abundant Numbers

An integer n > 1 is abundant if σ(n)/n ≥ 2, where σ(n) = Σ(d|n) d is the sum of divisors. The density of abundant numbers among positive integers is approximately 0.2476...

Covering Numbers

An integer n is covering if a distinct covering system of the integers can be formed using the divisors of n greater than one as moduli. A covering system is a collection of arithmetic progressions whose union covers all integers. The computational test uses covering index c'(n) ≥ 2, computed using generalize Bell numbers.

Output Format

FINAL RESULTS: Abundant Number Density Bounds
=============================================

COMPUTATION STATISTICS:
  Numbers processed:            12345678
  Total density accounted:      1.00000000002

DENSITY BOUNDS:
  Lower bound:       0.2474xxxxxxxxxxxxxx
  Upper bound:       0.2477xxxxxxxxxxxxxx
  Bound width:       0.0003xxxxxxxxxxxxxx

CONTRIBUTION BREAKDOWN:
  W₁ sum (early term):   0.123456789012345
  W₂ sum (power term):   0.098765432109876
  Given up branches:     0.025206340112346
  Lower bound (upper):   0.247428561234568

Note the Total density accounted should exceed 1 (it is designed to round up at every calculation.)

File Structure

├── AbunDens.cpp/.h          # Abundant number calculator
├── CovDens.cpp/.h           # Covering number calculator  
├── DensityMathFramework.cpp/.h  # Shared mathematical framework
├── CommonUtils.h            # Utility functions and constants
├── BellNums.h              # Precomputed Bell numbers for covering calculations
├── Makefile                # Build system
└── README.md               # This file

Citation

Please see the related paper for details on these calculations.

If you use these programs in research, please cite:

Nathan McNew and Jai Setty. On the density of covering numbers and abundant numbers.. 

License

This software is released under the GNU General Public License v3.0 (GPLv3). See the LICENSE file for full details.

This program 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 of the License, or (at your option) any later version.

This program 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.