Abundant odd numbers

Abundant odd numbers
You are encouraged to solve this task according to the task description, using any language you may know.

An Abundant number is a number n for which the   sum of divisors   σ(n) > 2n,
or,   equivalently,   the   sum of proper divisors   (or aliquot sum)       s(n) > n.

E.G.

12   is abundant, it has the proper divisors     1,2,3,4 & 6     which sum to   16   ( > 12 or n);
       or alternately,   has the sigma sum of   1,2,3,4,6 & 12   which sum to   28   ( > 24 or 2n).

Abundant numbers are common, though even abundant numbers seem to be much more common than odd abundant numbers.

To make things more interesting, this task is specifically about finding   odd abundant numbers.

Task

• Find and display here: at least the first 25 abundant odd numbers and either their proper divisor sum or sigma sum.
• Find and display here: the one thousandth abundant odd number and either its proper divisor sum or sigma sum.
• Find and display here: the first abundant odd number greater than one billion (10⁹) and either its proper divisor sum or sigma sum.

Reference

American Journal of Mathematics, Vol. 35, No. 4 (Oct., 1913), pp. 413-422 - Finiteness of the Odd Perfect and Primitive Abundant Numbers with n Distinct Prime Factors (LE Dickson)

ALGOL 68

<lang algol68>BEGIN

   # find some abundant odd numbers - numbers where the sum of the proper    #
   #                                  divisors is bigger than the number     #
   #                                  itself                                 #

   # returns the sum of the proper divisors of n                             #
   PROC divisor sum = ( INT n )INT:
   BEGIN
       INT sum := 1;
       FOR d FROM 2 TO ENTIER sqrt( n ) DO
           IF n MOD d = 0 THEN
               sum +:= d;
               IF INT other d := n OVER d;
                  other d /= d
               THEN
                   sum +:= other d
               FI
           FI
       OD;
       sum
   END # divisor sum # ;
   # find numbers required by the task                                       #
   BEGIN
       # first 25 odd abundant numbers                                       #
       INT odd number := 1;
       INT a count    := 0;
       INT d sum      := 0;
       print( ( "The first 25 abundant odd numbers:", newline ) );
       WHILE a count < 25 DO
           IF ( d sum := divisor sum( odd number ) ) > odd number THEN
               a count +:= 1;
               print( ( whole( odd number, -6 )
                      , " proper divisor sum: "
                      , whole( d sum, 0 )
                      , newline
                      )
                    )
           FI;
           odd number +:= 2
       OD;
       # 1000th odd abundant number                                          #
       WHILE a count < 1 000 DO
           IF ( d sum := divisor sum( odd number ) ) > odd number THEN
               a count := a count + 1
           FI;
           odd number +:= 2
       OD;
       print( ( "1000th abundant odd number:"
              , newline
              , "    "
              , whole( odd number - 2, 0 )
              , " proper divisor sum: "
              , whole( d sum, 0 )
              , newline
              )
            );
       # first odd abundant number > one billion                             #
       odd number := 1 000 000 001;
       BOOL found := FALSE;
       WHILE NOT found DO
           IF ( d sum := divisor sum( odd number ) ) > odd number THEN
               found  := TRUE;
               print( ( "First abundant odd number > 1 000 000 000:"
                      , newline
                      , "    "
                      , whole( odd number, 0 )
                      , " proper divisor sum: "
                      , whole( d sum, 0 )
                      , newline
                      )
                    )
           FI;
           odd number +:= 2
       OD
   END

END</lang>

Output:

The first 25 abundant odd numbers:
   945 proper divisor sum: 975
  1575 proper divisor sum: 1649
  2205 proper divisor sum: 2241
  2835 proper divisor sum: 2973
  3465 proper divisor sum: 4023
  4095 proper divisor sum: 4641
  4725 proper divisor sum: 5195
  5355 proper divisor sum: 5877
  5775 proper divisor sum: 6129
  5985 proper divisor sum: 6495
  6435 proper divisor sum: 6669
  6615 proper divisor sum: 7065
  6825 proper divisor sum: 7063
  7245 proper divisor sum: 7731
  7425 proper divisor sum: 7455
  7875 proper divisor sum: 8349
  8085 proper divisor sum: 8331
  8415 proper divisor sum: 8433
  8505 proper divisor sum: 8967
  8925 proper divisor sum: 8931
  9135 proper divisor sum: 9585
  9555 proper divisor sum: 9597
  9765 proper divisor sum: 10203
 10395 proper divisor sum: 12645
 11025 proper divisor sum: 11946
1000th abundant odd number:
    492975 proper divisor sum: 519361
First abundant odd number > 1 000 000 000:
    1000000575 proper divisor sum: 1083561009

BASIC256

Translation of: Visual Basic .NET

<lang BASIC256> numimpar = 1 contar = 0 sumaDiv = 0

function SumaDivisores(n) # Devuelve la suma de los divisores propios de n suma = 1 i = int(sqr(n))

for d = 2 to i if n % d = 0 then suma += d otroD = n \ d if otroD <> d Then suma += otroD end if Next d Return suma End Function

1. Encontrar los números requeridos por la tarea:

1. primeros 25 números abundantes impares

Print "Los primeros 25 números impares abundantes:" While contar < 25 sumaDiv = SumaDivisores(numimpar) If sumaDiv > numimpar Then contar += 1 Print numimpar & " suma divisoria adecuada: " & sumaDiv End If numimpar += 2 End While

1. 1000er número impar abundante

While contar < 1000 sumaDiv = SumaDivisores(numimpar) print sumaDiv & " " & contar If sumaDiv > numimpar Then contar += 1 numimpar += 2 End While Print Chr(10) & "1000º número impar abundante:" Print " " & (numimpar - 2) & " suma divisoria adecuada: " & sumaDiv

1. primer número impar abundante > mil millones (millardo)

numimpar = 1000000001 encontrado = False While Not encontrado sumaDiv = SumaDivisores(numimpar) If sumaDiv > numimpar Then encontrado = True Print Chr(10) & "Primer número impar abundante > 1 000 000 000:" Print " " & numimpar & " suma divisoria adecuada: " & sumaDiv End If numimpar += 2 End While End </lang>

C++

Translation of: Go

<lang cpp>#include <algorithm>

1. include <iostream>
2. include <numeric>
3. include <sstream>
4. include <vector>

std::vector<int> divisors(int n) {

   std::vector<int> divs{ 1 };
   std::vector<int> divs2;

   for (int i = 2; i*i <= n; i++) {
       if (n%i == 0) {
           int j = n / i;
           divs.push_back(i);
           if (i != j) {
               divs2.push_back(j);
           }
       }
   }
   std::copy(divs2.crbegin(), divs2.crend(), std::back_inserter(divs));

   return divs;

}

int sum(const std::vector<int>& divs) {

   return std::accumulate(divs.cbegin(), divs.cend(), 0);

}

std::string sumStr(const std::vector<int>& divs) {

   auto it = divs.cbegin();
   auto end = divs.cend();
   std::stringstream ss;

   if (it != end) {
       ss << *it;
       it = std::next(it);
   }
   while (it != end) {
       ss << " + " << *it;
       it = std::next(it);
   }

   return ss.str();

}

int abundantOdd(int searchFrom, int countFrom, int countTo, bool printOne) {

   int count = countFrom;
   int n = searchFrom;
   for (; count < countTo; n += 2) {
       auto divs = divisors(n);
       int tot = sum(divs);
       if (tot > n) {
           count++;
           if (printOne && count < countTo) {
               continue;
           }
           auto s = sumStr(divs);
           if (printOne) {
               printf("%d < %s = %d\n", n, s.c_str(), tot);
           } else {
               printf("%2d. %5d < %s = %d\n", count, n, s.c_str(), tot);
           }
       }
   }
   return n;

}

int main() {

   using namespace std;

   const int max = 25;
   cout << "The first " << max << " abundant odd primes are:\n";
   int n = abundantOdd(1, 0, 25, false);

   cout << "\nThe one thousandth abundant odd number is:\n";
   abundantOdd(n, 25, 1000, true);

   cout << "\nThe first abundant odd number above one billion is:\n";
   abundantOdd(1e9 + 1, 0, 1, true);

   return 0;

}</lang>

Output:

The first 25 abundant odd primes are:
 1.   945 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 105 + 135 + 189 + 315 = 975
 2.  1575 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 315 + 525 = 1649
 3.  2205 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 49 + 63 + 105 + 147 + 245 + 315 + 441 + 735 = 2241
 4.  2835 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 315 + 405 + 567 + 945 = 2973
 5.  3465 < 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 165 + 231 + 315 + 385 + 495 + 693 + 1155 = 4023
 6.  4095 < 1 + 3 + 5 + 7 + 9 + 13 + 15 + 21 + 35 + 39 + 45 + 63 + 65 + 91 + 105 + 117 + 195 + 273 + 315 + 455 + 585 + 819 + 1365 = 4641
 7.  4725 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 27 + 35 + 45 + 63 + 75 + 105 + 135 + 175 + 189 + 225 + 315 + 525 + 675 + 945 + 1575 = 5195
 8.  5355 < 1 + 3 + 5 + 7 + 9 + 15 + 17 + 21 + 35 + 45 + 51 + 63 + 85 + 105 + 119 + 153 + 255 + 315 + 357 + 595 + 765 + 1071 + 1785 = 5877
 9.  5775 < 1 + 3 + 5 + 7 + 11 + 15 + 21 + 25 + 33 + 35 + 55 + 75 + 77 + 105 + 165 + 175 + 231 + 275 + 385 + 525 + 825 + 1155 + 1925 = 6129
10.  5985 < 1 + 3 + 5 + 7 + 9 + 15 + 19 + 21 + 35 + 45 + 57 + 63 + 95 + 105 + 133 + 171 + 285 + 315 + 399 + 665 + 855 + 1197 + 1995 = 6495
11.  6435 < 1 + 3 + 5 + 9 + 11 + 13 + 15 + 33 + 39 + 45 + 55 + 65 + 99 + 117 + 143 + 165 + 195 + 429 + 495 + 585 + 715 + 1287 + 2145 = 6669
12.  6615 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 49 + 63 + 105 + 135 + 147 + 189 + 245 + 315 + 441 + 735 + 945 + 1323 + 2205 = 7065
13.  6825 < 1 + 3 + 5 + 7 + 13 + 15 + 21 + 25 + 35 + 39 + 65 + 75 + 91 + 105 + 175 + 195 + 273 + 325 + 455 + 525 + 975 + 1365 + 2275 = 7063
14.  7245 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 23 + 35 + 45 + 63 + 69 + 105 + 115 + 161 + 207 + 315 + 345 + 483 + 805 + 1035 + 1449 + 2415 = 7731
15.  7425 < 1 + 3 + 5 + 9 + 11 + 15 + 25 + 27 + 33 + 45 + 55 + 75 + 99 + 135 + 165 + 225 + 275 + 297 + 495 + 675 + 825 + 1485 + 2475 = 7455
16.  7875 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 125 + 175 + 225 + 315 + 375 + 525 + 875 + 1125 + 1575 + 2625 = 8349
17.  8085 < 1 + 3 + 5 + 7 + 11 + 15 + 21 + 33 + 35 + 49 + 55 + 77 + 105 + 147 + 165 + 231 + 245 + 385 + 539 + 735 + 1155 + 1617 + 2695 = 8331
18.  8415 < 1 + 3 + 5 + 9 + 11 + 15 + 17 + 33 + 45 + 51 + 55 + 85 + 99 + 153 + 165 + 187 + 255 + 495 + 561 + 765 + 935 + 1683 + 2805 = 8433
19.  8505 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 243 + 315 + 405 + 567 + 945 + 1215 + 1701 + 2835 = 8967
20.  8925 < 1 + 3 + 5 + 7 + 15 + 17 + 21 + 25 + 35 + 51 + 75 + 85 + 105 + 119 + 175 + 255 + 357 + 425 + 525 + 595 + 1275 + 1785 + 2975 = 8931
21.  9135 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 29 + 35 + 45 + 63 + 87 + 105 + 145 + 203 + 261 + 315 + 435 + 609 + 1015 + 1305 + 1827 + 3045 = 9585
22.  9555 < 1 + 3 + 5 + 7 + 13 + 15 + 21 + 35 + 39 + 49 + 65 + 91 + 105 + 147 + 195 + 245 + 273 + 455 + 637 + 735 + 1365 + 1911 + 3185 = 9597
23.  9765 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 31 + 35 + 45 + 63 + 93 + 105 + 155 + 217 + 279 + 315 + 465 + 651 + 1085 + 1395 + 1953 + 3255 = 10203
24. 10395 < 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 27 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 135 + 165 + 189 + 231 + 297 + 315 + 385 + 495 + 693 + 945 + 1155 + 1485 + 2079 + 3465 = 12645
25. 11025 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 = 11946

The one thousandth abundant odd number is:
492975 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 313 + 315 + 525 + 939 + 1565 + 1575 + 2191 + 2817 + 4695 + 6573 + 7825 + 10955 + 14085 + 19719 + 23475 + 32865 + 54775 + 70425 + 98595 + 164325 = 519361

The first abundant odd number above one billion is:
1000000575 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 + 11025 + 90703 + 272109 + 453515 + 634921 + 816327 + 1360545 + 1904763 + 2267575 + 3174605 + 4081635 + 4444447 + 5714289 + 6802725 + 9523815 + 13333341 + 15873025 + 20408175 + 22222235 + 28571445 + 40000023 + 47619075 + 66666705 + 111111175 + 142857225 + 200000115 + 333333525 = 1083561009

C#

<lang csharp>using static System.Console; using System.Collections.Generic; using System.Linq;

public static class AbundantOddNumbers {

   public static void Main() {
       WriteLine("First 25 abundant odd numbers:");
       foreach (var x in AbundantNumbers().Take(25)) WriteLine(x.Format());
       WriteLine();
       WriteLine($"The 1000th abundant odd number: {AbundantNumbers().ElementAt(999).Format()}");
       WriteLine();
       WriteLine($"First abundant odd number > 1b: {AbundantNumbers(1_000_000_001).First().Format()}");
   }

   static IEnumerable<(int n, int sum)> AbundantNumbers(int start = 3) =>
       start.UpBy(2).Select(n => (n, sum: n.DivisorSum())).Where(x => x.sum > x.n);

   static int DivisorSum(this int n) => 3.UpBy(2).TakeWhile(i => i * i <= n).Where(i => n % i == 0)
       .Select(i => (a:i, b:n/i)).Sum(p => p.a == p.b ? p.a : p.a + p.b) + 1;

   static IEnumerable<int> UpBy(this int n, int step) {
       for (int i = n; ; i+=step) yield return i;
   }

   static string Format(this (int n, int sum) pair) => $"{pair.n:N0} with sum {pair.sum:N0}";

}</lang>

Output:

First 25 abundant odd numbers:
945 with sum 975
1,575 with sum 1,649
2,205 with sum 2,241
2,835 with sum 2,973
3,465 with sum 4,023
4,095 with sum 4,641
4,725 with sum 5,195
5,355 with sum 5,877
5,775 with sum 6,129
5,985 with sum 6,495
6,435 with sum 6,669
6,615 with sum 7,065
6,825 with sum 7,063
7,245 with sum 7,731
7,425 with sum 7,455
7,875 with sum 8,349
8,085 with sum 8,331
8,415 with sum 8,433
8,505 with sum 8,967
8,925 with sum 8,931
9,135 with sum 9,585
9,555 with sum 9,597
9,765 with sum 10,203
10,395 with sum 12,645
11,025 with sum 11,946

The 1000th abundant odd number: 492,975 with sum 519,361

First abundant odd number > 1b: 1,000,000,575 with sum 1,083,561,009

D

Translation of: C++

<lang d>import std.stdio;

int[] divisors(int n) {

   import std.range;

   int[] divs = [1];
   int[] divs2;

   for (int i = 2; i * i <= n; i++) {
       if (n % i == 0) {
           int j = n / i;
           divs ~= i;
           if (i != j) {
               divs2 ~= j;
           }
       }
   }
   divs ~= retro(divs2).array;

   return divs;

}

int abundantOdd(int searchFrom, int countFrom, int countTo, bool printOne) {

   import std.algorithm.iteration;
   import std.array;
   import std.conv;

   int count = countFrom;
   int n = searchFrom;
   for (; count < countTo; n += 2) {
       auto divs = divisors(n);
       int tot = sum(divs);
       if (tot > n) {
           count++;
           if (printOne && count < countTo) {
               continue;
           }
           auto s = divs.map!(to!string).join(" + ");
           if (printOne) {
               writefln("%d < %s = %d", n, s, tot);
           } else {
               writefln("%2d. %5d < %s = %d", count, n, s, tot);
           }
       }
   }
   return n;

}

void main() {

   const int max = 25;
   writefln("The first %d abundant odd primes are:", max);
   int n = abundantOdd(1, 0, 25, false);

   writeln("\nThe one thousandth abundant odd number is:");
   abundantOdd(n, 25, 1000, true);

   writeln("\nThe first abundant odd number above one billion is:");
   abundantOdd(cast(int)(1e9 + 1), 0, 1, true);

}</lang>

Output:

The first 25 abundant odd primes are:
 1.   945 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 105 + 135 + 189 + 315 = 975
 2.  1575 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 315 + 525 = 1649
 3.  2205 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 49 + 63 + 105 + 147 + 245 + 315 + 441 + 735 = 2241
 4.  2835 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 315 + 405 + 567 + 945 = 2973
 5.  3465 < 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 165 + 231 + 315 + 385 + 495 + 693 + 1155 = 4023
 6.  4095 < 1 + 3 + 5 + 7 + 9 + 13 + 15 + 21 + 35 + 39 + 45 + 63 + 65 + 91 + 105 + 117 + 195 + 273 + 315 + 455 + 585 + 819 + 1365 = 4641
 7.  4725 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 27 + 35 + 45 + 63 + 75 + 105 + 135 + 175 + 189 + 225 + 315 + 525 + 675 + 945 + 1575 = 5195
 8.  5355 < 1 + 3 + 5 + 7 + 9 + 15 + 17 + 21 + 35 + 45 + 51 + 63 + 85 + 105 + 119 + 153 + 255 + 315 + 357 + 595 + 765 + 1071 + 1785 = 5877
 9.  5775 < 1 + 3 + 5 + 7 + 11 + 15 + 21 + 25 + 33 + 35 + 55 + 75 + 77 + 105 + 165 + 175 + 231 + 275 + 385 + 525 + 825 + 1155 + 1925 = 6129
10.  5985 < 1 + 3 + 5 + 7 + 9 + 15 + 19 + 21 + 35 + 45 + 57 + 63 + 95 + 105 + 133 + 171 + 285 + 315 + 399 + 665 + 855 + 1197 + 1995 = 6495
11.  6435 < 1 + 3 + 5 + 9 + 11 + 13 + 15 + 33 + 39 + 45 + 55 + 65 + 99 + 117 + 143 + 165 + 195 + 429 + 495 + 585 + 715 + 1287 + 2145 = 6669
12.  6615 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 49 + 63 + 105 + 135 + 147 + 189 + 245 + 315 + 441 + 735 + 945 + 1323 + 2205 = 7065
13.  6825 < 1 + 3 + 5 + 7 + 13 + 15 + 21 + 25 + 35 + 39 + 65 + 75 + 91 + 105 + 175 + 195 + 273 + 325 + 455 + 525 + 975 + 1365 + 2275 = 7063
14.  7245 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 23 + 35 + 45 + 63 + 69 + 105 + 115 + 161 + 207 + 315 + 345 + 483 + 805 + 1035 + 1449 + 2415 = 7731
15.  7425 < 1 + 3 + 5 + 9 + 11 + 15 + 25 + 27 + 33 + 45 + 55 + 75 + 99 + 135 + 165 + 225 + 275 + 297 + 495 + 675 + 825 + 1485 + 2475 = 7455
16.  7875 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 125 + 175 + 225 + 315 + 375 + 525 + 875 + 1125 + 1575 + 2625 = 8349
17.  8085 < 1 + 3 + 5 + 7 + 11 + 15 + 21 + 33 + 35 + 49 + 55 + 77 + 105 + 147 + 165 + 231 + 245 + 385 + 539 + 735 + 1155 + 1617 + 2695 = 8331
18.  8415 < 1 + 3 + 5 + 9 + 11 + 15 + 17 + 33 + 45 + 51 + 55 + 85 + 99 + 153 + 165 + 187 + 255 + 495 + 561 + 765 + 935 + 1683 + 2805 = 8433
19.  8505 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 243 + 315 + 405 + 567 + 945 + 1215 + 1701 + 2835 = 8967
20.  8925 < 1 + 3 + 5 + 7 + 15 + 17 + 21 + 25 + 35 + 51 + 75 + 85 + 105 + 119 + 175 + 255 + 357 + 425 + 525 + 595 + 1275 + 1785 + 2975 = 8931
21.  9135 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 29 + 35 + 45 + 63 + 87 + 105 + 145 + 203 + 261 + 315 + 435 + 609 + 1015 + 1305 + 1827 + 3045 = 9585
22.  9555 < 1 + 3 + 5 + 7 + 13 + 15 + 21 + 35 + 39 + 49 + 65 + 91 + 105 + 147 + 195 + 245 + 273 + 455 + 637 + 735 + 1365 + 1911 + 3185 = 9597
23.  9765 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 31 + 35 + 45 + 63 + 93 + 105 + 155 + 217 + 279 + 315 + 465 + 651 + 1085 + 1395 + 1953 + 3255 = 10203
24. 10395 < 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 27 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 135 + 165 + 189 + 231 + 297 + 315 + 385 + 495 + 693 + 945 + 1155 + 1485 + 2079 + 3465 = 12645
25. 11025 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 = 11946

The one thousandth abundant odd number is:
492975 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 313 + 315 + 525 + 939 + 1565 + 1575 + 2191 + 2817 + 4695 + 6573 + 7825 + 10955 + 14085 + 19719 + 23475 + 32865 + 54775 + 70425 + 98595 + 164325 = 519361

The first abundant odd number above one billion is:
1000000575 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 + 11025 + 90703 + 272109 + 453515 + 634921 + 816327 + 1360545 + 1904763 + 2267575 + 3174605 + 4081635 + 4444447 + 5714289 + 6802725 + 9523815 + 13333341 + 15873025 + 20408175 + 22222235 + 28571445 + 40000023 + 47619075 + 66666705 + 111111175 + 142857225 + 200000115 + 333333525 = 1083561009

Factor

<lang factor>USING: arrays formatting io kernel lists lists.lazy math math.primes.factors sequences tools.memory.private ; IN: rosetta-code.abundant-odd-numbers

σ ( n -- sum ) divisors sum ;

abundant? ( n -- ? ) [ σ ] [ 2 * ] bi > ;

abundant-odds-from ( n -- list )

   dup even? [ 1 + ] when
   [ 2 + ] lfrom-by [ abundant? ] lfilter ;

first25 ( -- seq ) 25 1 abundant-odds-from ltake list>array ;

1,000th ( -- n ) 1 abundant-odds-from 999 [ cdr ] times car ;

first>10^9 ( -- n ) 1,000,000,001 abundant-odds-from car ;

GENERIC: show ( obj -- ) M: integer show dup σ [ commas ] bi@ "%-6s σ = %s\n" printf ; M: array show [ show ] each ;

abundant-odd-numbers-demo ( -- )

   first25 "First 25 abundant odd numbers:"
   1,000th "1,000th abundant odd number:"
   first>10^9 "First abundant odd number > one billion:"
   [ print show nl ] 2tri@ ;

MAIN: abundant-odd-numbers-demo</lang>

Output:

First 25 abundant odd numbers:
945    σ = 1,920
1,575  σ = 3,224
2,205  σ = 4,446
2,835  σ = 5,808
3,465  σ = 7,488
4,095  σ = 8,736
4,725  σ = 9,920
5,355  σ = 11,232
5,775  σ = 11,904
5,985  σ = 12,480
6,435  σ = 13,104
6,615  σ = 13,680
6,825  σ = 13,888
7,245  σ = 14,976
7,425  σ = 14,880
7,875  σ = 16,224
8,085  σ = 16,416
8,415  σ = 16,848
8,505  σ = 17,472
8,925  σ = 17,856
9,135  σ = 18,720
9,555  σ = 19,152
9,765  σ = 19,968
10,395 σ = 23,040
11,025 σ = 22,971

1,000th abundant odd number:
492,975 σ = 1,012,336

First abundant odd number > one billion:
1,000,000,575 σ = 2,083,561,584

FreeBASIC

Translation of: Visual Basic .NET

<lang freebasic> Declare Function SumaDivisores(n As Integer) As Integer

Dim numimpar As Integer = 1 Dim contar As Integer = 0 Dim sumaDiv As Integer = 0

Function SumaDivisores(n As Integer) As Integer

   ' Devuelve la suma de los divisores propios de n
   Dim suma As Integer = 1
   Dim As Integer d, otroD
   
   For d = 2 To Cint(Sqr(n))
       If n Mod d = 0 Then
           suma += d
           otroD = n \ d
           If otroD <> d Then suma += otroD
       End If
   Next d
   Return suma

End Function

' Encontrar los números requeridos por la tarea:

' primeros 25 números abundantes impares Print "Los primeros 25 números impares abundantes:" Do While contar < 25

   sumaDiv = SumaDivisores(numimpar)
   If sumaDiv > numimpar Then
       contar += 1
       Print using "######"; numimpar;
       Print " suma divisoria adecuada: " & sumaDiv
   End If
   numimpar += 2

Loop

' 1000er número impar abundante Do While contar < 1000

   sumaDiv = SumaDivisores(numimpar)
   If sumaDiv > numimpar Then contar += 1
   numimpar += 2

Loop Print Chr(10) & "1000º número impar abundante:" Print " " & (numimpar - 2) & " suma divisoria adecuada: " & sumaDiv

' primer número impar abundante > mil millones (millardo) numimpar = 1000000001 Dim encontrado As Boolean = False Do While Not encontrado

   sumaDiv = SumaDivisores(numimpar)
   If sumaDiv > numimpar Then
       encontrado = True
       Print Chr(10) & "Primer número impar abundante > 1 000 000 000:"
       Print "    " & numimpar & " suma divisoria adecuada: " & sumaDiv
   End If
   numimpar += 2

Loop End </lang>

Output:

Los primeros 25 números impares abundantes:
   945 suma divisoria adecuada: 975
  1575 suma divisoria adecuada: 1649
  2205 suma divisoria adecuada: 2241
  2835 suma divisoria adecuada: 2973
  3465 suma divisoria adecuada: 4023
  4095 suma divisoria adecuada: 4641
  4725 suma divisoria adecuada: 5195
  5355 suma divisoria adecuada: 5877
  5775 suma divisoria adecuada: 6129
  5985 suma divisoria adecuada: 6495
  6435 suma divisoria adecuada: 6669
  6615 suma divisoria adecuada: 7065
  6825 suma divisoria adecuada: 7063
  7245 suma divisoria adecuada: 7731
  7425 suma divisoria adecuada: 7455
  7875 suma divisoria adecuada: 8349
  8085 suma divisoria adecuada: 8331
  8415 suma divisoria adecuada: 8433
  8505 suma divisoria adecuada: 8967
  8925 suma divisoria adecuada: 8931
  9135 suma divisoria adecuada: 9585
  9555 suma divisoria adecuada: 9597
  9765 suma divisoria adecuada: 10203
 10395 suma divisoria adecuada: 12645
 11025 suma divisoria adecuada: 11946

1000º número impar abundante:
    492975 suma divisoria adecuada: 519361

Primer número impar abundante > 1 000 000 000:
    1000000575 suma divisoria adecuada: 1083561009

Go

<lang go>package main

import (

   "fmt"
   "strconv"

)

func divisors(n int) []int {

   divs := []int{1}
   divs2 := []int{}
   for i := 2; i*i <= n; i++ {
       if n%i == 0 {
           j := n / i
           divs = append(divs, i)
           if i != j {
               divs2 = append(divs2, j)
           }
       }
   }
   for i := len(divs2) - 1; i >= 0; i-- {
       divs = append(divs, divs2[i])
   }
   return divs

}

func sum(divs []int) int {

   tot := 0
   for _, div := range divs {
       tot += div
   }
   return tot

}

func sumStr(divs []int) string {

   s := ""
   for _, div := range divs {
       s += strconv.Itoa(div) + " + "
   }
   return s[0 : len(s)-3]

}

func abundantOdd(searchFrom, countFrom, countTo int, printOne bool) int {

   count := countFrom
   n := searchFrom
   for ; count < countTo; n += 2 {
       divs := divisors(n)
       if tot := sum(divs); tot > n {
           count++
           if printOne && count < countTo {
               continue
           } 
           s := sumStr(divs)
           if !printOne {
               fmt.Printf("%2d. %5d < %s = %d\n", count, n, s, tot)
           } else {
               fmt.Printf("%d < %s = %d\n", n, s, tot)
           }
       }
   }
   return n

}

func main() {

   const max = 25
   fmt.Println("The first", max, "abundant odd numbers are:")
   n := abundantOdd(1, 0, 25, false)

   fmt.Println("\nThe one thousandth abundant odd number is:")
   abundantOdd(n, 25, 1000, true)

   fmt.Println("\nThe first abundant odd number above one billion is:")
   abundantOdd(1e9+1, 0, 1, true)

}</lang>

Output:

The first 25 abundant odd numbers are:
 1.   945 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 105 + 135 + 189 + 315 = 975
 2.  1575 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 315 + 525 = 1649
 3.  2205 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 49 + 63 + 105 + 147 + 245 + 315 + 441 + 735 = 2241
 4.  2835 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 315 + 405 + 567 + 945 = 2973
 5.  3465 < 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 165 + 231 + 315 + 385 + 495 + 693 + 1155 = 4023
 6.  4095 < 1 + 3 + 5 + 7 + 9 + 13 + 15 + 21 + 35 + 39 + 45 + 63 + 65 + 91 + 105 + 117 + 195 + 273 + 315 + 455 + 585 + 819 + 1365 = 4641
 7.  4725 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 27 + 35 + 45 + 63 + 75 + 105 + 135 + 175 + 189 + 225 + 315 + 525 + 675 + 945 + 1575 = 5195
 8.  5355 < 1 + 3 + 5 + 7 + 9 + 15 + 17 + 21 + 35 + 45 + 51 + 63 + 85 + 105 + 119 + 153 + 255 + 315 + 357 + 595 + 765 + 1071 + 1785 = 5877
 9.  5775 < 1 + 3 + 5 + 7 + 11 + 15 + 21 + 25 + 33 + 35 + 55 + 75 + 77 + 105 + 165 + 175 + 231 + 275 + 385 + 525 + 825 + 1155 + 1925 = 6129
10.  5985 < 1 + 3 + 5 + 7 + 9 + 15 + 19 + 21 + 35 + 45 + 57 + 63 + 95 + 105 + 133 + 171 + 285 + 315 + 399 + 665 + 855 + 1197 + 1995 = 6495
11.  6435 < 1 + 3 + 5 + 9 + 11 + 13 + 15 + 33 + 39 + 45 + 55 + 65 + 99 + 117 + 143 + 165 + 195 + 429 + 495 + 585 + 715 + 1287 + 2145 = 6669
12.  6615 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 49 + 63 + 105 + 135 + 147 + 189 + 245 + 315 + 441 + 735 + 945 + 1323 + 2205 = 7065
13.  6825 < 1 + 3 + 5 + 7 + 13 + 15 + 21 + 25 + 35 + 39 + 65 + 75 + 91 + 105 + 175 + 195 + 273 + 325 + 455 + 525 + 975 + 1365 + 2275 = 7063
14.  7245 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 23 + 35 + 45 + 63 + 69 + 105 + 115 + 161 + 207 + 315 + 345 + 483 + 805 + 1035 + 1449 + 2415 = 7731
15.  7425 < 1 + 3 + 5 + 9 + 11 + 15 + 25 + 27 + 33 + 45 + 55 + 75 + 99 + 135 + 165 + 225 + 275 + 297 + 495 + 675 + 825 + 1485 + 2475 = 7455
16.  7875 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 125 + 175 + 225 + 315 + 375 + 525 + 875 + 1125 + 1575 + 2625 = 8349
17.  8085 < 1 + 3 + 5 + 7 + 11 + 15 + 21 + 33 + 35 + 49 + 55 + 77 + 105 + 147 + 165 + 231 + 245 + 385 + 539 + 735 + 1155 + 1617 + 2695 = 8331
18.  8415 < 1 + 3 + 5 + 9 + 11 + 15 + 17 + 33 + 45 + 51 + 55 + 85 + 99 + 153 + 165 + 187 + 255 + 495 + 561 + 765 + 935 + 1683 + 2805 = 8433
19.  8505 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 243 + 315 + 405 + 567 + 945 + 1215 + 1701 + 2835 = 8967
20.  8925 < 1 + 3 + 5 + 7 + 15 + 17 + 21 + 25 + 35 + 51 + 75 + 85 + 105 + 119 + 175 + 255 + 357 + 425 + 525 + 595 + 1275 + 1785 + 2975 = 8931
21.  9135 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 29 + 35 + 45 + 63 + 87 + 105 + 145 + 203 + 261 + 315 + 435 + 609 + 1015 + 1305 + 1827 + 3045 = 9585
22.  9555 < 1 + 3 + 5 + 7 + 13 + 15 + 21 + 35 + 39 + 49 + 65 + 91 + 105 + 147 + 195 + 245 + 273 + 455 + 637 + 735 + 1365 + 1911 + 3185 = 9597
23.  9765 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 31 + 35 + 45 + 63 + 93 + 105 + 155 + 217 + 279 + 315 + 465 + 651 + 1085 + 1395 + 1953 + 3255 = 10203
24. 10395 < 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 27 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 135 + 165 + 189 + 231 + 297 + 315 + 385 + 495 + 693 + 945 + 1155 + 1485 + 2079 + 3465 = 12645
25. 11025 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 = 11946

The one thousandth abundant odd number is:
492975 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 313 + 315 + 525 + 939 + 1565 + 1575 + 2191 + 2817 + 4695 + 6573 + 7825 + 10955 + 14085 + 19719 + 23475 + 32865 + 54775 + 70425 + 98595 + 164325 = 519361

The first abundant odd number above one billion is:
1000000575 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 + 11025 + 90703 + 272109 + 453515 + 634921 + 816327 + 1360545 + 1904763 + 2267575 + 3174605 + 4081635 + 4444447 + 5714289 + 6802725 + 9523815 + 13333341 + 15873025 + 20408175 + 22222235 + 28571445 + 40000023 + 47619075 + 66666705 + 111111175 + 142857225 + 200000115 + 333333525 = 1083561009

J

   NB. https://www.math.upenn.edu/~deturck/m170/wk3/lecture/sumdiv.html
   s=: ([: */ [: ((<:@:(^ >:)/) % <:@:{.) __&q:)&>

   assert 6045 -: s 1800

   aliquot_sum=: -~ s

   abundant=: < aliquot_sum

   Filter=: (#~`)(`:6)

   A=: abundant Filter 1 2 p. i. 260000  NB. a batch of abundant odd numbers

   # A   NB. more than 1000, it's enough.
1054

   NB. the first odd abundant numbers
   (,: aliquot_sum) 26 {. A
945 1575 2205 2835 3465 4095 4725 5355 5775 5985 6435 6615 6825 7245 7425 7875 8085 8415 8505 8925 9135 9555  9765 10395 11025 11655
975 1649 2241 2973 4023 4641 5195 5877 6129 6495 6669 7065 7063 7731 7455 8349 8331 8433 8967 8931 9585 9597 10203 12645 11946 12057

   NB. the one thousandth abundant odd number
   (,: aliquot_sum) 999 { A
492975
519361


   k=: adverb def '1000 * m'
   1x k k k
1000000000

   abundant Filter (1x k k k) + 1 2x p. i. 10x k
1000000575 1000001475 1000001625 1000001835 1000002465 1000003095 1000003725 1000004355 1000004775 1000004985 1000005435 1000005615 1000005825 1000006245 1000006425 1000006875 1000007505 1000008765 1000009395 1000010025 1000010655 1000011285 1000011705 100...

   (,: aliquot_sum) {. abundant Filter (1x k k k) + 1 2x p. i. 10x k
1000000575
1083561009

Julia

<lang julia>using Primes

function propfact(n)

   f = [one(n)]
   for (p, x) in factor(n)
       f = reduce(vcat, [f*p^i for i in 1:x], init=f)
   end
   pop!(f)
   f

end

isabundant(n) = sum(propfact(n)) > n prettyprintfactors(n) = (a = propfact(n); println("$n has proper divisors $a, these sum to $(sum(a))."))

function oddabundantsfrom(startingint, needed, nprint=0)

   n = isodd(startingint) ? startingint : startingint + 1
   count = 0
   while count < needed
       if isabundant(n)
           if nprint == 0
               prettyprintfactors(n)
           elseif nprint == count + 1
               prettyprintfactors(n)
               break
           end
           count += 1
       end
       n += 2
   end

end

println("First 25 abundant odd numbers:") oddabundantsfrom(2, 25)

println("The thousandth abundant odd number:") oddabundantsfrom(2, 1001, 1000)

println("The first abundant odd number greater than one billion:") oddabundantsfrom(1000000000, 1)

</lang>

Output:

First 25 abundant odd numbers:
945 has proper divisors [1, 3, 9, 27, 5, 15, 45, 135, 7, 21, 63, 189, 35, 105, 315], these sum to 975.
1575 has proper divisors [1, 3, 9, 5, 15, 45, 25, 75, 225, 7, 21, 63, 35, 105, 315, 175, 525], these sum to 1649.
2205 has proper divisors [1, 3, 9, 5, 15, 45, 7, 21, 63, 35, 105, 315, 49, 147, 441, 245, 735], these sum to 2241.
2835 has proper divisors [1, 3, 9, 27, 81, 5, 15, 45, 135, 405, 7, 21, 63, 189, 567, 35, 105, 315, 945], these sum to 2973.
3465 has proper divisors [1, 3, 9, 5, 15, 45, 7, 21, 63, 35, 105, 315, 11, 33, 99, 55, 165, 495, 77, 231, 693, 385, 1155], these sum to 4023.
4095 has proper divisors [1, 3, 9, 5, 15, 45, 7, 21, 63, 35, 105, 315, 13, 39, 117, 65, 195, 585, 91, 273, 819, 455, 1365], these sum to 4641.
4725 has proper divisors [1, 3, 9, 27, 5, 15, 45, 135, 25, 75, 225, 675, 7, 21, 63, 189, 35, 105, 315, 945, 175, 525, 1575], these sum to 5195.
5355 has proper divisors [1, 3, 9, 5, 15, 45, 7, 21, 63, 35, 105, 315, 17, 51, 153, 85, 255, 765, 119, 357, 1071, 595, 1785], these sum to 5877.
5775 has proper divisors [1, 3, 5, 15, 25, 75, 7, 21, 35, 105, 175, 525, 11, 33, 55, 165, 275, 825, 77, 231, 385, 1155, 1925], these sum to 6129.
5985 has proper divisors [1, 3, 9, 5, 15, 45, 7, 21, 63, 35, 105, 315, 19, 57, 171, 95, 285, 855, 133, 399, 1197, 665, 1995], these sum to 6495.
6435 has proper divisors [1, 3, 9, 5, 15, 45, 11, 33, 99, 55, 165, 495, 13, 39, 117, 65, 195, 585, 143, 429, 1287, 715, 2145], these sum to 6669.
6615 has proper divisors [1, 3, 9, 27, 5, 15, 45, 135, 7, 21, 63, 189, 35, 105, 315, 945, 49, 147, 441, 1323, 245, 735, 2205], these sum to 7065.
6825 has proper divisors [1, 3, 5, 15, 25, 75, 7, 21, 35, 105, 175, 525, 13, 39, 65, 195, 325, 975, 91, 273, 455, 1365, 2275], these sum to 7063.
7245 has proper divisors [1, 3, 9, 5, 15, 45, 7, 21, 63, 35, 105, 315, 23, 69, 207, 115, 345, 1035, 161, 483, 1449, 805, 2415], these sum to 7731.
7425 has proper divisors [1, 3, 9, 27, 5, 15, 45, 135, 25, 75, 225, 675, 11, 33, 99, 297, 55, 165, 495, 1485, 275, 825, 2475], these sum to 7455.
7875 has proper divisors [1, 3, 9, 5, 15, 45, 25, 75, 225, 125, 375, 1125, 7, 21, 63, 35, 105, 315, 175, 525, 1575, 875, 2625], these sum to 8349.
8085 has proper divisors [1, 3, 5, 15, 7, 21, 35, 105, 49, 147, 245, 735, 11, 33, 55, 165, 77, 231, 385, 1155, 539, 1617, 2695], these sum to 8331.
8415 has proper divisors [1, 3, 9, 5, 15, 45, 11, 33, 99, 55, 165, 495, 17, 51, 153, 85, 255, 765, 187, 561, 1683, 935, 2805], these sum to 8433.
8505 has proper divisors [1, 3, 9, 27, 81, 243, 5, 15, 45, 135, 405, 1215, 7, 21, 63, 189, 567, 1701, 35, 105, 315, 945, 2835], these sum to 8967.
8925 has proper divisors [1, 3, 5, 15, 25, 75, 7, 21, 35, 105, 175, 525, 17, 51, 85, 255, 425, 1275, 119, 357, 595, 1785, 2975], these sum to 8931.
9135 has proper divisors [1, 3, 9, 5, 15, 45, 7, 21, 63, 35, 105, 315, 29, 87, 261, 145, 435, 1305, 203, 609, 1827, 1015, 3045], these sum to 9585.
9555 has proper divisors [1, 3, 5, 15, 7, 21, 35, 105, 49, 147, 245, 735, 13, 39, 65, 195, 91, 273, 455, 1365, 637, 1911, 3185], these sum to 9597.
9765 has proper divisors [1, 3, 9, 5, 15, 45, 7, 21, 63, 35, 105, 315, 31, 93, 279, 155, 465, 1395, 217, 651, 1953, 1085, 3255], these sum to 10203.
10395 has proper divisors [1, 3, 9, 27, 5, 15, 45, 135, 7, 21, 63, 189, 35, 105, 315, 945, 11, 33, 99, 297, 55, 165, 495, 1485, 77, 231, 693, 2079, 385, 1155, 3465], these sum to 12645.
11025 has proper divisors [1, 3, 9, 5, 15, 45, 25, 75, 225, 7, 21, 63, 35, 105, 315, 175, 525, 1575, 49, 147, 441, 245, 735, 2205, 1225, 3675], these sum to 11946.
The thousandth abundant odd number:
492975 has proper divisors [1, 3, 9, 5, 15, 45, 25, 75, 225, 7, 21, 63, 35, 105, 315, 175, 525, 1575, 313, 939, 2817, 1565, 4695, 14085, 7825, 23475, 70425, 2191, 6573, 19719, 10955, 32865, 98595, 54775, 164325], these sum to 519361.
The first abundant odd number greater than one billion:
1000000575 has proper divisors [1, 3, 9, 5, 15, 45, 25, 75, 225, 7, 21, 63, 35, 105, 315, 175, 525, 1575, 49, 147, 441, 245, 735, 2205, 1225, 3675, 11025, 90703, 272109, 816327, 453515, 1360545, 4081635, 2267575, 6802725, 20408175, 634921, 1904763, 5714289, 3174605, 9523815, 28571445, 15873025, 47619075, 142857225, 4444447, 13333341, 40000023, 22222235, 66666705, 200000115, 111111175, 333333525], these sum to 1083561009.

Kotlin

Translation of: D

<lang scala>fun divisors(n: Int): List<Int> {

   val divs = mutableListOf(1)
   val divs2 = mutableListOf<Int>()

   var i = 2
   while (i * i <= n) {
       if (n % i == 0) {
           val j = n / i
           divs.add(i)
           if (i != j) {
               divs2.add(j)
           }
       }
       i++
   }

   divs.addAll(divs2.reversed())

   return divs

}

fun abundantOdd(searchFrom: Int, countFrom: Int, countTo: Int, printOne: Boolean): Int {

   var count = countFrom
   var n = searchFrom

   while (count < countTo) {
       val divs = divisors(n)
       val tot = divs.sum()
       if (tot > n) {
           count++
           if (!printOne || count >= countTo) {
               val s = divs.joinToString(" + ")
               if (printOne) {
                   println("$n < $s = $tot")
               } else {
                   println("%2d. %5d < %s = %d".format(count, n, s, tot))
               }
           }
       }

       n += 2
   }

   return n

}

fun main() {

   val max = 25
   println("The first $max abundant primes are:")
   val n = abundantOdd(1, 0, 25, false)

   println("\nThe one thousandth abundant odd number is:")
   abundantOdd(n, 25, 1000, true)

   println("\nThe first abundant odd number above one billion is:")
   abundantOdd((1e9 + 1).toInt(), 0, 1, true)

}</lang>

Output:

The first 25 abundant primes are:
 1.   945 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 105 + 135 + 189 + 315 = 975
 2.  1575 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 315 + 525 = 1649
 3.  2205 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 49 + 63 + 105 + 147 + 245 + 315 + 441 + 735 = 2241
 4.  2835 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 315 + 405 + 567 + 945 = 2973
 5.  3465 < 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 165 + 231 + 315 + 385 + 495 + 693 + 1155 = 4023
 6.  4095 < 1 + 3 + 5 + 7 + 9 + 13 + 15 + 21 + 35 + 39 + 45 + 63 + 65 + 91 + 105 + 117 + 195 + 273 + 315 + 455 + 585 + 819 + 1365 = 4641
 7.  4725 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 27 + 35 + 45 + 63 + 75 + 105 + 135 + 175 + 189 + 225 + 315 + 525 + 675 + 945 + 1575 = 5195
 8.  5355 < 1 + 3 + 5 + 7 + 9 + 15 + 17 + 21 + 35 + 45 + 51 + 63 + 85 + 105 + 119 + 153 + 255 + 315 + 357 + 595 + 765 + 1071 + 1785 = 5877
 9.  5775 < 1 + 3 + 5 + 7 + 11 + 15 + 21 + 25 + 33 + 35 + 55 + 75 + 77 + 105 + 165 + 175 + 231 + 275 + 385 + 525 + 825 + 1155 + 1925 = 6129
10.  5985 < 1 + 3 + 5 + 7 + 9 + 15 + 19 + 21 + 35 + 45 + 57 + 63 + 95 + 105 + 133 + 171 + 285 + 315 + 399 + 665 + 855 + 1197 + 1995 = 6495
11.  6435 < 1 + 3 + 5 + 9 + 11 + 13 + 15 + 33 + 39 + 45 + 55 + 65 + 99 + 117 + 143 + 165 + 195 + 429 + 495 + 585 + 715 + 1287 + 2145 = 6669
12.  6615 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 49 + 63 + 105 + 135 + 147 + 189 + 245 + 315 + 441 + 735 + 945 + 1323 + 2205 = 7065
13.  6825 < 1 + 3 + 5 + 7 + 13 + 15 + 21 + 25 + 35 + 39 + 65 + 75 + 91 + 105 + 175 + 195 + 273 + 325 + 455 + 525 + 975 + 1365 + 2275 = 7063
14.  7245 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 23 + 35 + 45 + 63 + 69 + 105 + 115 + 161 + 207 + 315 + 345 + 483 + 805 + 1035 + 1449 + 2415 = 7731
15.  7425 < 1 + 3 + 5 + 9 + 11 + 15 + 25 + 27 + 33 + 45 + 55 + 75 + 99 + 135 + 165 + 225 + 275 + 297 + 495 + 675 + 825 + 1485 + 2475 = 7455
16.  7875 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 125 + 175 + 225 + 315 + 375 + 525 + 875 + 1125 + 1575 + 2625 = 8349
17.  8085 < 1 + 3 + 5 + 7 + 11 + 15 + 21 + 33 + 35 + 49 + 55 + 77 + 105 + 147 + 165 + 231 + 245 + 385 + 539 + 735 + 1155 + 1617 + 2695 = 8331
18.  8415 < 1 + 3 + 5 + 9 + 11 + 15 + 17 + 33 + 45 + 51 + 55 + 85 + 99 + 153 + 165 + 187 + 255 + 495 + 561 + 765 + 935 + 1683 + 2805 = 8433
19.  8505 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 243 + 315 + 405 + 567 + 945 + 1215 + 1701 + 2835 = 8967
20.  8925 < 1 + 3 + 5 + 7 + 15 + 17 + 21 + 25 + 35 + 51 + 75 + 85 + 105 + 119 + 175 + 255 + 357 + 425 + 525 + 595 + 1275 + 1785 + 2975 = 8931
21.  9135 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 29 + 35 + 45 + 63 + 87 + 105 + 145 + 203 + 261 + 315 + 435 + 609 + 1015 + 1305 + 1827 + 3045 = 9585
22.  9555 < 1 + 3 + 5 + 7 + 13 + 15 + 21 + 35 + 39 + 49 + 65 + 91 + 105 + 147 + 195 + 245 + 273 + 455 + 637 + 735 + 1365 + 1911 + 3185 = 9597
23.  9765 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 31 + 35 + 45 + 63 + 93 + 105 + 155 + 217 + 279 + 315 + 465 + 651 + 1085 + 1395 + 1953 + 3255 = 10203
24. 10395 < 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 27 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 135 + 165 + 189 + 231 + 297 + 315 + 385 + 495 + 693 + 945 + 1155 + 1485 + 2079 + 3465 = 12645
25. 11025 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 = 11946

The one thousandth abundant odd number is:
492975 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 313 + 315 + 525 + 939 + 1565 + 1575 + 2191 + 2817 + 4695 + 6573 + 7825 + 10955 + 14085 + 19719 + 23475 + 32865 + 54775 + 70425 + 98595 + 164325 = 519361

The first abundant odd number above one billion is:
1000000575 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 + 11025 + 90703 + 272109 + 453515 + 634921 + 816327 + 1360545 + 1904763 + 2267575 + 3174605 + 4081635 + 4444447 + 5714289 + 6802725 + 9523815 + 13333341 + 15873025 + 20408175 + 22222235 + 28571445 + 40000023 + 47619075 + 66666705 + 111111175 + 142857225 + 200000115 + 333333525 = 1083561009

Perl

Translation of: Perl 6

Library: ntheory

<lang perl>use strict; use warnings; use feature 'say'; use ntheory qw/divisor_sum divisors/;

sub odd_abundants {

   my($start,$count) = @_;
   my $n = int(( $start + 2 ) / 3);
   $n   += 1 if 0 == $n / 2;
   $n   *= 3;
   my @out;
   while (@out < $count) {
       $n += 6;
       next unless (my $ds = divisor_sum($n)) > 2*$n;
       my @d = divisors($n);
       push @out, sprintf "%6d: divisor sum: %s = %d", $n, join(' + ', @d[0..@d-2]), $ds-$n;
   }
   @out;

}

say 'First 25 abundant odd numbers:'; say for odd_abundants(1, 25); say "\nOne thousandth abundant odd number:\n", (odd_abundants(1, 1000))[999]; say "\nFirst abundant odd number above one billion:\n", odd_abundants(999_999_999, 1);</lang>

Output:

First 25 abundant odd numbers:
   945: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 105 + 135 + 189 + 315 = 975
  1575: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 315 + 525 = 1649
  2205: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 49 + 63 + 105 + 147 + 245 + 315 + 441 + 735 = 2241
  2835: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 315 + 405 + 567 + 945 = 2973
  3465: divisor sum: 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 165 + 231 + 315 + 385 + 495 + 693 + 1155 = 4023
  4095: divisor sum: 1 + 3 + 5 + 7 + 9 + 13 + 15 + 21 + 35 + 39 + 45 + 63 + 65 + 91 + 105 + 117 + 195 + 273 + 315 + 455 + 585 + 819 + 1365 = 4641
  4725: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 27 + 35 + 45 + 63 + 75 + 105 + 135 + 175 + 189 + 225 + 315 + 525 + 675 + 945 + 1575 = 5195
  5355: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 17 + 21 + 35 + 45 + 51 + 63 + 85 + 105 + 119 + 153 + 255 + 315 + 357 + 595 + 765 + 1071 + 1785 = 5877
  5775: divisor sum: 1 + 3 + 5 + 7 + 11 + 15 + 21 + 25 + 33 + 35 + 55 + 75 + 77 + 105 + 165 + 175 + 231 + 275 + 385 + 525 + 825 + 1155 + 1925 = 6129
  5985: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 19 + 21 + 35 + 45 + 57 + 63 + 95 + 105 + 133 + 171 + 285 + 315 + 399 + 665 + 855 + 1197 + 1995 = 6495
  6435: divisor sum: 1 + 3 + 5 + 9 + 11 + 13 + 15 + 33 + 39 + 45 + 55 + 65 + 99 + 117 + 143 + 165 + 195 + 429 + 495 + 585 + 715 + 1287 + 2145 = 6669
  6615: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 49 + 63 + 105 + 135 + 147 + 189 + 245 + 315 + 441 + 735 + 945 + 1323 + 2205 = 7065
  6825: divisor sum: 1 + 3 + 5 + 7 + 13 + 15 + 21 + 25 + 35 + 39 + 65 + 75 + 91 + 105 + 175 + 195 + 273 + 325 + 455 + 525 + 975 + 1365 + 2275 = 7063
  7245: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 23 + 35 + 45 + 63 + 69 + 105 + 115 + 161 + 207 + 315 + 345 + 483 + 805 + 1035 + 1449 + 2415 = 7731
  7425: divisor sum: 1 + 3 + 5 + 9 + 11 + 15 + 25 + 27 + 33 + 45 + 55 + 75 + 99 + 135 + 165 + 225 + 275 + 297 + 495 + 675 + 825 + 1485 + 2475 = 7455
  7875: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 125 + 175 + 225 + 315 + 375 + 525 + 875 + 1125 + 1575 + 2625 = 8349
  8085: divisor sum: 1 + 3 + 5 + 7 + 11 + 15 + 21 + 33 + 35 + 49 + 55 + 77 + 105 + 147 + 165 + 231 + 245 + 385 + 539 + 735 + 1155 + 1617 + 2695 = 8331
  8415: divisor sum: 1 + 3 + 5 + 9 + 11 + 15 + 17 + 33 + 45 + 51 + 55 + 85 + 99 + 153 + 165 + 187 + 255 + 495 + 561 + 765 + 935 + 1683 + 2805 = 8433
  8505: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 243 + 315 + 405 + 567 + 945 + 1215 + 1701 + 2835 = 8967
  8925: divisor sum: 1 + 3 + 5 + 7 + 15 + 17 + 21 + 25 + 35 + 51 + 75 + 85 + 105 + 119 + 175 + 255 + 357 + 425 + 525 + 595 + 1275 + 1785 + 2975 = 8931
  9135: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 29 + 35 + 45 + 63 + 87 + 105 + 145 + 203 + 261 + 315 + 435 + 609 + 1015 + 1305 + 1827 + 3045 = 9585
  9555: divisor sum: 1 + 3 + 5 + 7 + 13 + 15 + 21 + 35 + 39 + 49 + 65 + 91 + 105 + 147 + 195 + 245 + 273 + 455 + 637 + 735 + 1365 + 1911 + 3185 = 9597
  9765: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 31 + 35 + 45 + 63 + 93 + 105 + 155 + 217 + 279 + 315 + 465 + 651 + 1085 + 1395 + 1953 + 3255 = 10203
 10395: divisor sum: 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 27 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 135 + 165 + 189 + 231 + 297 + 315 + 385 + 495 + 693 + 945 + 1155 + 1485 + 2079 + 3465 = 12645
 11025: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 = 11946

One thousandth abundant odd number:
492975: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 313 + 315 + 525 + 939 + 1565 + 1575 + 2191 + 2817 + 4695 + 6573 + 7825 + 10955 + 14085 + 19719 + 23475 + 32865 + 54775 + 70425 + 98595 + 164325 = 519361

First abundant odd number above one billion:
1000000575: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 + 11025 + 90703 + 272109 + 453515 + 634921 + 816327 + 1360545 + 1904763 + 2267575 + 3174605 + 4081635 + 4444447 + 5714289 + 6802725 + 9523815 + 13333341 + 15873025 + 20408175 + 22222235 + 28571445 + 40000023 + 47619075 + 66666705 + 111111175 + 142857225 + 200000115 + 333333525 = 1083561009

Perl 6

Works with: Rakudo version 2019.03

<lang perl6>sub odd-abundant (\x) {

   my @l = x.is-prime ?? 1 !! flat
   1, (3 .. x.sqrt.floor).map: -> \d {
        next unless d +& 1;
        my \y = x div d;
        next if y * d !== x;
        d !== y ?? (d, y) !! d
   };
   @l.sum > x ?? @l.sort !! Empty;

}

sub odd-abundants (Int :$start-at is copy) {

   $start-at = ( $start-at + 2 ) div 3;
   $start-at += $start-at %% 2;
   $start-at *= 3;
   ($start-at, *+6 ... *).hyper.map: {
       next unless my $oa = .&odd-abundant;
       sprintf "%6d: divisor sum: {$oa.join: ' + '} = {$oa.sum}", $_
   }

}

put 'First 25 abundant odd numbers:'; .put for odd-abundants( :start-at(1) )[^25];

put "\nOne thousandth abundant odd number:\n" ~ odd-abundants( :start-at(1) )[999] ~

"\n\nFirst abundant odd number above one billion:\n" ~ odd-abundants( :start-at(1_000_000_000) ).head;</lang>

Output:

First 25 abundant odd numbers:
   945: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 105 + 135 + 189 + 315 = 975
  1575: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 315 + 525 = 1649
  2205: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 49 + 63 + 105 + 147 + 245 + 315 + 441 + 735 = 2241
  2835: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 315 + 405 + 567 + 945 = 2973
  3465: divisor sum: 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 165 + 231 + 315 + 385 + 495 + 693 + 1155 = 4023
  4095: divisor sum: 1 + 3 + 5 + 7 + 9 + 13 + 15 + 21 + 35 + 39 + 45 + 63 + 65 + 91 + 105 + 117 + 195 + 273 + 315 + 455 + 585 + 819 + 1365 = 4641
  4725: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 27 + 35 + 45 + 63 + 75 + 105 + 135 + 175 + 189 + 225 + 315 + 525 + 675 + 945 + 1575 = 5195
  5355: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 17 + 21 + 35 + 45 + 51 + 63 + 85 + 105 + 119 + 153 + 255 + 315 + 357 + 595 + 765 + 1071 + 1785 = 5877
  5775: divisor sum: 1 + 3 + 5 + 7 + 11 + 15 + 21 + 25 + 33 + 35 + 55 + 75 + 77 + 105 + 165 + 175 + 231 + 275 + 385 + 525 + 825 + 1155 + 1925 = 6129
  5985: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 19 + 21 + 35 + 45 + 57 + 63 + 95 + 105 + 133 + 171 + 285 + 315 + 399 + 665 + 855 + 1197 + 1995 = 6495
  6435: divisor sum: 1 + 3 + 5 + 9 + 11 + 13 + 15 + 33 + 39 + 45 + 55 + 65 + 99 + 117 + 143 + 165 + 195 + 429 + 495 + 585 + 715 + 1287 + 2145 = 6669
  6615: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 49 + 63 + 105 + 135 + 147 + 189 + 245 + 315 + 441 + 735 + 945 + 1323 + 2205 = 7065
  6825: divisor sum: 1 + 3 + 5 + 7 + 13 + 15 + 21 + 25 + 35 + 39 + 65 + 75 + 91 + 105 + 175 + 195 + 273 + 325 + 455 + 525 + 975 + 1365 + 2275 = 7063
  7245: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 23 + 35 + 45 + 63 + 69 + 105 + 115 + 161 + 207 + 315 + 345 + 483 + 805 + 1035 + 1449 + 2415 = 7731
  7425: divisor sum: 1 + 3 + 5 + 9 + 11 + 15 + 25 + 27 + 33 + 45 + 55 + 75 + 99 + 135 + 165 + 225 + 275 + 297 + 495 + 675 + 825 + 1485 + 2475 = 7455
  7875: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 125 + 175 + 225 + 315 + 375 + 525 + 875 + 1125 + 1575 + 2625 = 8349
  8085: divisor sum: 1 + 3 + 5 + 7 + 11 + 15 + 21 + 33 + 35 + 49 + 55 + 77 + 105 + 147 + 165 + 231 + 245 + 385 + 539 + 735 + 1155 + 1617 + 2695 = 8331
  8415: divisor sum: 1 + 3 + 5 + 9 + 11 + 15 + 17 + 33 + 45 + 51 + 55 + 85 + 99 + 153 + 165 + 187 + 255 + 495 + 561 + 765 + 935 + 1683 + 2805 = 8433
  8505: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 243 + 315 + 405 + 567 + 945 + 1215 + 1701 + 2835 = 8967
  8925: divisor sum: 1 + 3 + 5 + 7 + 15 + 17 + 21 + 25 + 35 + 51 + 75 + 85 + 105 + 119 + 175 + 255 + 357 + 425 + 525 + 595 + 1275 + 1785 + 2975 = 8931
  9135: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 29 + 35 + 45 + 63 + 87 + 105 + 145 + 203 + 261 + 315 + 435 + 609 + 1015 + 1305 + 1827 + 3045 = 9585
  9555: divisor sum: 1 + 3 + 5 + 7 + 13 + 15 + 21 + 35 + 39 + 49 + 65 + 91 + 105 + 147 + 195 + 245 + 273 + 455 + 637 + 735 + 1365 + 1911 + 3185 = 9597
  9765: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 31 + 35 + 45 + 63 + 93 + 105 + 155 + 217 + 279 + 315 + 465 + 651 + 1085 + 1395 + 1953 + 3255 = 10203
 10395: divisor sum: 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 27 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 135 + 165 + 189 + 231 + 297 + 315 + 385 + 495 + 693 + 945 + 1155 + 1485 + 2079 + 3465 = 12645
 11025: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 = 11946

One thousandth abundant odd number:
492975: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 313 + 315 + 525 + 939 + 1565 + 1575 + 2191 + 2817 + 4695 + 6573 + 7825 + 10955 + 14085 + 19719 + 23475 + 32865 + 54775 + 70425 + 98595 + 164325 = 519361

First abundant odd number above one billion:
1000000575: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 + 11025 + 90703 + 272109 + 453515 + 634921 + 816327 + 1360545 + 1904763 + 2267575 + 3174605 + 4081635 + 4444447 + 5714289 + 6802725 + 9523815 + 13333341 + 15873025 + 20408175 + 22222235 + 28571445 + 40000023 + 47619075 + 66666705 + 111111175 + 142857225 + 200000115 + 333333525 = 1083561009

Phix

<lang Phix>function abundantOdd(integer n, done, lim, bool printAll)

   while done<lim do
       atom tot = sum(factors(n,-1))
       if tot>n then
           done += 1
           if printAll or done=lim then
               string ln = iff(printAll?sprintf("%2d. ",done):"")
               printf(1,"%s%,6d (proper sum:%,d)\n",{ln,n,tot})
           end if
       end if
       n += 2
   end while
   printf(1,"\n")
   return n

end function printf(1,"The first 25 abundant odd numbers are:\n") integer n = abundantOdd(1, 0, 25, true) printf(1,"The one thousandth abundant odd number is:") {} = abundantOdd(n, 25, 1000, false) printf(1,"The first abundant odd number above one billion is:") {} = abundantOdd(1e9+1, 0, 1, false)</lang>

Output:

The first 25 abundant odd numbers are:
 1.    945 (proper sum:975)
 2.  1,575 (proper sum:1,649)
 3.  2,205 (proper sum:2,241)
 4.  2,835 (proper sum:2,973)
 5.  3,465 (proper sum:4,023)
 6.  4,095 (proper sum:4,641)
 7.  4,725 (proper sum:5,195)
 8.  5,355 (proper sum:5,877)
 9.  5,775 (proper sum:6,129)
10.  5,985 (proper sum:6,495)
11.  6,435 (proper sum:6,669)
12.  6,615 (proper sum:7,065)
13.  6,825 (proper sum:7,063)
14.  7,245 (proper sum:7,731)
15.  7,425 (proper sum:7,455)
16.  7,875 (proper sum:8,349)
17.  8,085 (proper sum:8,331)
18.  8,415 (proper sum:8,433)
19.  8,505 (proper sum:8,967)
20.  8,925 (proper sum:8,931)
21.  9,135 (proper sum:9,585)
22.  9,555 (proper sum:9,597)
23.  9,765 (proper sum:10,203)
24. 10,395 (proper sum:12,645)
25. 11,025 (proper sum:11,946)

The one thousandth abundant odd number is:492,975 (proper sum:519,361)

The first abundant odd number above one billion is:1,000,000,575 (proper sum:1,083,561,009)

Python

Translation of: BASIC256

<lang Python>

1. !/usr/bin/python

numimpar = 1 contar = 0 sumaDiv = 0

from math import sqrt

def SumaDivisores(n):

   # Devuelve la suma de los divisores propios de n
   suma = 1
   i = int(sqrt(n))
       
   for d in range (2, i):
       if n % d == 0:
           suma += d
           otroD = n // d
           if otroD != d:
               suma += otroD
   return suma

1. los números requeridos por la tarea:

1. primeros 25 números abundantes impares

print ("Los primeros 25 números impares abundantes:") while contar < 25:

   sumaDiv = SumaDivisores(numimpar)
   if sumaDiv > numimpar:
       contar += 1
       print("{0:5} suma divisoria adecuada: {1}". format(numimpar,sumaDiv))
   numimpar += 2

1. número impar abundante

while contar < 1000:

   sumaDiv = SumaDivisores(numimpar)
   if sumaDiv > numimpar:
       contar += 1
   numimpar += 2

print ("\n1000º número impar abundante:") print (f' {numimpar - 2} suma divisoria adecuada: {sumaDiv}')

1. primer número impar abundante > mil millones (millardo)

numimpar = 1000000001 encontrado = False while not encontrado:

   sumaDiv = SumaDivisores(numimpar)
   if sumaDiv > numimpar:
       encontrado = True
       print ("\nPrimer número impar abundante > 1 000 000 000:")
       print (f'    {numimpar} suma divisoria adecuada: {sumaDiv}')
   numimpar += 2

</lang>

Output:

Los primeros 25 números impares abundantes:
  945 suma divisoria adecuada: 975
 1575 suma divisoria adecuada: 1649
 2205 suma divisoria adecuada: 2241
 2835 suma divisoria adecuada: 2973
 3465 suma divisoria adecuada: 4023
 4095 suma divisoria adecuada: 4513
 4725 suma divisoria adecuada: 5195
 5355 suma divisoria adecuada: 5877
 5775 suma divisoria adecuada: 5977
 5985 suma divisoria adecuada: 6495
 6435 suma divisoria adecuada: 6669
 6615 suma divisoria adecuada: 7065
 6825 suma divisoria adecuada: 7063
 7245 suma divisoria adecuada: 7731
 7425 suma divisoria adecuada: 7455
 7875 suma divisoria adecuada: 8349
 8085 suma divisoria adecuada: 8331
 8415 suma divisoria adecuada: 8433
 8505 suma divisoria adecuada: 8967
 8925 suma divisoria adecuada: 8931
 9135 suma divisoria adecuada: 9585
 9555 suma divisoria adecuada: 9597
 9765 suma divisoria adecuada: 10203
10395 suma divisoria adecuada: 12645
11025 suma divisoria adecuada: 11841

1000º número impar abundante:
    492975 suma divisoria adecuada: 519361

Primer número impar abundante > 1 000 000 000:
    1000000575 suma divisoria adecuada: 1083561009

REXX

A wee bit of coding was added to add commas (because of the larger numbers) as well as alignment of the output. <lang rexx>/*REXX pgm displays abundant odd numbers: 1st 25, one─thousandth, first > 1 billion. */ parse arg Nlow Nuno Novr . /*obtain optional arguments from the CL*/ if Nlow== | Nlow=="," then Nlow= 25 /*Not specified? Then use the default.*/ if Nuno== | Nuno=="," then Nuno= 1000 /* " " " " " " */ if Novr== | Novr=="," then Novr= 1000000000 /* " " " " " " */ numeric digits max(9, length(Novr) ) /*ensure enough decimal digits for // */ @= 'odd abundant number' /*variable for annotating the output. */

1. = 0 /*count of odd abundant numbers so far.*/

     do j=3  by 2  until #>=Nlow;   $= sigO(j)  /*get the  sigma  for an odd integer.  */
     if $<=j  then iterate                      /*sigma  ≤  J ?    Then ignore it.     */
     #= # + 1                                   /*bump the counter for abundant odd #'s*/
     say rt(th(#))   @    'is:'rt(commas(j), 8)    rt('sigma=')    rt(commas($), 9)
     end  /*j*/

say

1. = 0 /*count of odd abundant numbers so far.*/

     do j=3  by 2;                  $= sigO(j)  /*get the  sigma  for an odd integer.  */
     if $<=j    then iterate                    /*sigma  ≤  J ?    Then ignore it.     */
     #= # + 1                                   /*bump the counter for abundant odd #'s*/
     if #<Nuno  then iterate                    /*Odd abundant# count<Nuno?  Then skip.*/
     say rt(th(#))   @    'is:'rt(commas(j), 8)    rt('sigma=')    rt(commas($), 9)
     leave                                      /*we're finished displaying NUNOth num.*/
     end  /*j*/

say

     do j=1+Novr%2*2  by 2;         $= sigO(j)  /*get sigma for an odd integer > Novr. */
     if $<=j    then iterate                    /*sigma  ≤  J ?    Then ignore it.     */
     say rt(th(1))   @  'over'  commas(Novr)  "is: "   commas(j)  rt('sigma=')  commas($)
     leave                                      /*we're finished displaying NOVRth num.*/
     end  /*j*/

exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ commas:parse arg _; do c_=length(_)-3 to 1 by -3; _=insert(',', _, c_); end; return _ rt: procedure; parse arg #,len; if len== then len= 20; return right(#, len) th: parse arg th; return th||word('th st nd rd',1+(th//10)*(th//100%10\==1)*(th//10<4)) /*──────────────────────────────────────────────────────────────────────────────────────*/ sigO: procedure; parse arg x; s= 1 /*sigma for odd integers. ___*/

            do k=3  by 2  while k*k<x           /*divide by all odd integers up to √ x */
            if x//k==0  then  s= s + k +  x%k   /*add the two divisors to (sigma) sum. */
            end   /*k*/                         /*                                  ___*/
      if k*k==x  then  return s + k             /*Was  X  a square?    If so, add  √ x */
                       return s                 /*return (sigma) sum of the divisors.  */</lang>

output   when using the default input:

                 1st odd abundant number is:     945               sigma=       975
                 2nd odd abundant number is:   1,575               sigma=     1,649
                 3rd odd abundant number is:   2,205               sigma=     2,241
                 4th odd abundant number is:   2,835               sigma=     2,973
                 5th odd abundant number is:   3,465               sigma=     4,023
                 6th odd abundant number is:   4,095               sigma=     4,641
                 7th odd abundant number is:   4,725               sigma=     5,195
                 8th odd abundant number is:   5,355               sigma=     5,877
                 9th odd abundant number is:   5,775               sigma=     6,129
                10th odd abundant number is:   5,985               sigma=     6,495
                11th odd abundant number is:   6,435               sigma=     6,669
                12th odd abundant number is:   6,615               sigma=     7,065
                13th odd abundant number is:   6,825               sigma=     7,063
                14th odd abundant number is:   7,245               sigma=     7,731
                15th odd abundant number is:   7,425               sigma=     7,455
                16th odd abundant number is:   7,875               sigma=     8,349
                17th odd abundant number is:   8,085               sigma=     8,331
                18th odd abundant number is:   8,415               sigma=     8,433
                19th odd abundant number is:   8,505               sigma=     8,967
                20th odd abundant number is:   8,925               sigma=     8,931
                21st odd abundant number is:   9,135               sigma=     9,585
                22nd odd abundant number is:   9,555               sigma=     9,597
                23rd odd abundant number is:   9,765               sigma=    10,203
                24th odd abundant number is:  10,395               sigma=    12,645
                25th odd abundant number is:  11,025               sigma=    11,946

              1000th odd abundant number is: 492,975               sigma=   519,361

                 1st odd abundant number over 1,000,000,000 is:  1,000,000,575               sigma= 1,083,561,009

Ring

<lang ring>

1. Project: Anbundant odd numbers

max = 100000000 limit = 25 nr = 0 m = 1 check = 0 index = 0 see "working..." + nl see "wait for done..." + nl while true

     check = 0
     if m%2 = 1
        nice(m)
     ok
     if check = 1
        nr = nr + 1
     ok
     if nr = max
        exit
     ok
     m = m + 1

end see "done..." + nl

func nice(n)

    check = 0
    nArray = []
    for i = 1 to n - 1
        if n % i = 0
           add(nArray,i)
        ok
    next
    sum = 0
    for p = 1 to len(nArray)
        sum = sum + nArray[p]
    next
    if sum > n
       check = 1
       index = index + 1
       if index < limit + 1
          showArray(n,nArray,sum,index)
       ok
       if index = 100
          see "One thousandth abundant odd number:" + nl
          showArray2(n,nArray,sum,index)
       ok
       if index = 100000000
          see "First abundant odd number above one billion:" + nl
          showArray2(n,nArray,sum,index)
       ok
    ok

func showArray(n,nArray,sum,index)

       see "" + index + ". " + string(n) + ": divisor sum: " 
       for m = 1 to len(nArray)
           if m < len(nArray)
              see string(nArray[m]) + " + "
           else
              see string(nArray[m]) + " = " + string(sum) + nl + nl
           ok
       next

func showArray2(n,nArray,sum,index)

       see "" + index + ". " + string(n) + ": divisor sum: " + 
       see string(nArray[m]) + " = " + string(sum) + nl + nl

</lang>

working...
wait for done...
1. 945: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 105 + 135 + 189 + 315 = 975

2. 1575: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 315 + 525 = 1649

3. 2205: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 49 + 63 + 105 + 147 + 245 + 315 + 441 + 735 = 2241

4. 2835: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 315 + 405 + 567 + 945 = 2973

5. 3465: divisor sum: 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 165 + 231 + 315 + 385 + 495 + 693 + 1155 = 4023

6. 4095: divisor sum: 1 + 3 + 5 + 7 + 9 + 13 + 15 + 21 + 35 + 39 + 45 + 63 + 65 + 91 + 105 + 117 + 195 + 273 + 315 + 455 + 585 + 819 + 1365 = 4641

7. 4725: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 27 + 35 + 45 + 63 + 75 + 105 + 135 + 175 + 189 + 225 + 315 + 525 + 675 + 945 + 1575 = 5195

8. 5355: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 17 + 21 + 35 + 45 + 51 + 63 + 85 + 105 + 119 + 153 + 255 + 315 + 357 + 595 + 765 + 1071 + 1785 = 5877

9. 5775: divisor sum: 1 + 3 + 5 + 7 + 11 + 15 + 21 + 25 + 33 + 35 + 55 + 75 + 77 + 105 + 165 + 175 + 231 + 275 + 385 + 525 + 825 + 1155 + 1925 = 6129

10. 5985: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 19 + 21 + 35 + 45 + 57 + 63 + 95 + 105 + 133 + 171 + 285 + 315 + 399 + 665 + 855 + 1197 + 1995 = 6495

11. 6435: divisor sum: 1 + 3 + 5 + 9 + 11 + 13 + 15 + 33 + 39 + 45 + 55 + 65 + 99 + 117 + 143 + 165 + 195 + 429 + 495 + 585 + 715 + 1287 + 2145 = 6669

12. 6615: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 49 + 63 + 105 + 135 + 147 + 189 + 245 + 315 + 441 + 735 + 945 + 1323 + 2205 = 7065

13. 6825: divisor sum: 1 + 3 + 5 + 7 + 13 + 15 + 21 + 25 + 35 + 39 + 65 + 75 + 91 + 105 + 175 + 195 + 273 + 325 + 455 + 525 + 975 + 1365 + 2275 = 7063

14. 7245: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 23 + 35 + 45 + 63 + 69 + 105 + 115 + 161 + 207 + 315 + 345 + 483 + 805 + 1035 + 1449 + 2415 = 7731

15. 7425: divisor sum: 1 + 3 + 5 + 9 + 11 + 15 + 25 + 27 + 33 + 45 + 55 + 75 + 99 + 135 + 165 + 225 + 275 + 297 + 495 + 675 + 825 + 1485 + 2475 = 7455

16. 7875: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 125 + 175 + 225 + 315 + 375 + 525 + 875 + 1125 + 1575 + 2625 = 8349

17. 8085: divisor sum: 1 + 3 + 5 + 7 + 11 + 15 + 21 + 33 + 35 + 49 + 55 + 77 + 105 + 147 + 165 + 231 + 245 + 385 + 539 + 735 + 1155 + 1617 + 2695 = 8331

18. 8415: divisor sum: 1 + 3 + 5 + 9 + 11 + 15 + 17 + 33 + 45 + 51 + 55 + 85 + 99 + 153 + 165 + 187 + 255 + 495 + 561 + 765 + 935 + 1683 + 2805 = 8433

19. 8505: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 243 + 315 + 405 + 567 + 945 + 1215 + 1701 + 2835 = 8967

20. 8925: divisor sum: 1 + 3 + 5 + 7 + 15 + 17 + 21 + 25 + 35 + 51 + 75 + 85 + 105 + 119 + 175 + 255 + 357 + 425 + 525 + 595 + 1275 + 1785 + 2975 = 8931

21. 9135: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 29 + 35 + 45 + 63 + 87 + 105 + 145 + 203 + 261 + 315 + 435 + 609 + 1015 + 1305 + 1827 + 3045 = 9585

22. 9555: divisor sum: 1 + 3 + 5 + 7 + 13 + 15 + 21 + 35 + 39 + 49 + 65 + 91 + 105 + 147 + 195 + 245 + 273 + 455 + 637 + 735 + 1365 + 1911 + 3185 = 9597

23. 9765: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 31 + 35 + 45 + 63 + 93 + 105 + 155 + 217 + 279 + 315 + 465 + 651 + 1085 + 1395 + 1953 + 3255 = 10203

24. 10395: divisor sum: 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 27 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 135 + 165 + 189 + 231 + 297 + 315 + 385 + 495 + 693 + 945 + 1155 + 1485 + 2079 + 3465 = 12645

25. 11025: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 = 11946

One thousandth abundant odd number:
1000. 492975: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 313 + 315 + 525 + 939 + 1565 + 1575 + 2191 + 2817 + 4695 + 6573 + 7825 + 10955 + 14085 + 19719 + 23475 + 32865 + 54775 + 70425 + 98595 + 164325 = 519361

First abundant odd number above one billion:
100000000. 1000000575: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 + 11025 + 90703 + 272109 + 453515 + 634921 + 816327 + 1360545 + 1904763 + 2267575 + 3174605 + 4081635 + 4444447 + 5714289 + 6802725 + 9523815 + 13333341 + 15873025 + 20408175 + 22222235 + 28571445 + 40000023 + 47619075 + 66666705 + 111111175 + 142857225 + 200000115 + 333333525 = 1083561009
done...

Ruby

proper_divisors method taken from http://rosettacode.org/wiki/Proper_divisors#Ruby <lang ruby>require "prime"

class Integer

 def proper_divisors
   return [] if self == 1
   primes = prime_division.flat_map{|prime, freq| [prime] * freq}
   (1...primes.size).each_with_object([1]) do |n, res|
     primes.combination(n).map{|combi| res << combi.inject(:*)}
   end.flatten.uniq
 end

end

def generator_odd_abundants(from=1)

 from += 1 if from.even?
 Enumerator.new do |y|
   from.step(nil, 2) do |n|
     sum = n.proper_divisors.sum
     y << [n, sum] if sum > n
   end
 end

end

generator_odd_abundants.take(25).each{|n, sum| puts "#{n} with sum #{sum}" } puts "\n%d with sum %#d" % generator_odd_abundants.take(1000).last puts "\n%d with sum %#d" % generator_odd_abundants(1_000_000_000).next </lang>

Scala

Translation of: D

<lang scala>import scala.collection.mutable.ListBuffer

object Abundant {

 def divisors(n: Int): ListBuffer[Int] = {
   val divs = new ListBuffer[Int]
   divs.append(1)

   val divs2 = new ListBuffer[Int]
   var i = 2

   while (i * i <= n) {
     if (n % i == 0) {
       val j = n / i
       divs.append(i)
       if (i != j) {
         divs2.append(j)
       }
     }
     i += 1
   }

   divs.appendAll(divs2.reverse)
   divs
 }

 def abundantOdd(searchFrom: Int, countFrom: Int, countTo: Int, printOne: Boolean): Int = {
   var count = countFrom
   var n = searchFrom
   while (count < countTo) {
     val divs = divisors(n)
     val tot = divs.sum
     if (tot > n) {
       count += 1
       if (!printOne || !(count < countTo)) {
         val s = divs.map(a => a.toString).mkString(" + ")
         if (printOne) {
           printf("%d < %s = %d\n", n, s, tot)
         } else {
           printf("%2d. %5d < %s = %d\n", count, n, s, tot)
         }
       }
     }
     n += 2
   }

   n
 }

 def main(args: Array[String]): Unit = {
   val max = 25
   printf("The first %d abundant odd primes are:\n", max)
   val n = abundantOdd(1, 0, max, printOne = false)

   printf("\nThe one thousandth abundant odd number is:\n")
   abundantOdd(n, 25, 1000, printOne = true)

   printf("\nThe first abundant odd number above one billion is:\n")
   abundantOdd((1e9 + 1).intValue(), 0, 1, printOne = true)
 }

}</lang>

Output:

The first 25 abundant odd primes are:
 1.   945 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 105 + 135 + 189 + 315 = 975
 2.  1575 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 315 + 525 = 1649
 3.  2205 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 49 + 63 + 105 + 147 + 245 + 315 + 441 + 735 = 2241
 4.  2835 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 315 + 405 + 567 + 945 = 2973
 5.  3465 < 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 165 + 231 + 315 + 385 + 495 + 693 + 1155 = 4023
 6.  4095 < 1 + 3 + 5 + 7 + 9 + 13 + 15 + 21 + 35 + 39 + 45 + 63 + 65 + 91 + 105 + 117 + 195 + 273 + 315 + 455 + 585 + 819 + 1365 = 4641
 7.  4725 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 27 + 35 + 45 + 63 + 75 + 105 + 135 + 175 + 189 + 225 + 315 + 525 + 675 + 945 + 1575 = 5195
 8.  5355 < 1 + 3 + 5 + 7 + 9 + 15 + 17 + 21 + 35 + 45 + 51 + 63 + 85 + 105 + 119 + 153 + 255 + 315 + 357 + 595 + 765 + 1071 + 1785 = 5877
 9.  5775 < 1 + 3 + 5 + 7 + 11 + 15 + 21 + 25 + 33 + 35 + 55 + 75 + 77 + 105 + 165 + 175 + 231 + 275 + 385 + 525 + 825 + 1155 + 1925 = 6129
10.  5985 < 1 + 3 + 5 + 7 + 9 + 15 + 19 + 21 + 35 + 45 + 57 + 63 + 95 + 105 + 133 + 171 + 285 + 315 + 399 + 665 + 855 + 1197 + 1995 = 6495
11.  6435 < 1 + 3 + 5 + 9 + 11 + 13 + 15 + 33 + 39 + 45 + 55 + 65 + 99 + 117 + 143 + 165 + 195 + 429 + 495 + 585 + 715 + 1287 + 2145 = 6669
12.  6615 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 49 + 63 + 105 + 135 + 147 + 189 + 245 + 315 + 441 + 735 + 945 + 1323 + 2205 = 7065
13.  6825 < 1 + 3 + 5 + 7 + 13 + 15 + 21 + 25 + 35 + 39 + 65 + 75 + 91 + 105 + 175 + 195 + 273 + 325 + 455 + 525 + 975 + 1365 + 2275 = 7063
14.  7245 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 23 + 35 + 45 + 63 + 69 + 105 + 115 + 161 + 207 + 315 + 345 + 483 + 805 + 1035 + 1449 + 2415 = 7731
15.  7425 < 1 + 3 + 5 + 9 + 11 + 15 + 25 + 27 + 33 + 45 + 55 + 75 + 99 + 135 + 165 + 225 + 275 + 297 + 495 + 675 + 825 + 1485 + 2475 = 7455
16.  7875 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 125 + 175 + 225 + 315 + 375 + 525 + 875 + 1125 + 1575 + 2625 = 8349
17.  8085 < 1 + 3 + 5 + 7 + 11 + 15 + 21 + 33 + 35 + 49 + 55 + 77 + 105 + 147 + 165 + 231 + 245 + 385 + 539 + 735 + 1155 + 1617 + 2695 = 8331
18.  8415 < 1 + 3 + 5 + 9 + 11 + 15 + 17 + 33 + 45 + 51 + 55 + 85 + 99 + 153 + 165 + 187 + 255 + 495 + 561 + 765 + 935 + 1683 + 2805 = 8433
19.  8505 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 243 + 315 + 405 + 567 + 945 + 1215 + 1701 + 2835 = 8967
20.  8925 < 1 + 3 + 5 + 7 + 15 + 17 + 21 + 25 + 35 + 51 + 75 + 85 + 105 + 119 + 175 + 255 + 357 + 425 + 525 + 595 + 1275 + 1785 + 2975 = 8931
21.  9135 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 29 + 35 + 45 + 63 + 87 + 105 + 145 + 203 + 261 + 315 + 435 + 609 + 1015 + 1305 + 1827 + 3045 = 9585
22.  9555 < 1 + 3 + 5 + 7 + 13 + 15 + 21 + 35 + 39 + 49 + 65 + 91 + 105 + 147 + 195 + 245 + 273 + 455 + 637 + 735 + 1365 + 1911 + 3185 = 9597
23.  9765 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 31 + 35 + 45 + 63 + 93 + 105 + 155 + 217 + 279 + 315 + 465 + 651 + 1085 + 1395 + 1953 + 3255 = 10203
24. 10395 < 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 27 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 135 + 165 + 189 + 231 + 297 + 315 + 385 + 495 + 693 + 945 + 1155 + 1485 + 2079 + 3465 = 12645
25. 11025 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 = 11946

The one thousandth abundant odd number is:
492975 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 313 + 315 + 525 + 939 + 1565 + 1575 + 2191 + 2817 + 4695 + 6573 + 7825 + 10955 + 14085 + 19719 + 23475 + 32865 + 54775 + 70425 + 98595 + 164325 = 519361

The first abundant odd number above one billion is:
1000000575 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 + 11025 + 90703 + 272109 + 453515 + 634921 + 816327 + 1360545 + 1904763 + 2267575 + 3174605 + 4081635 + 4444447 + 5714289 + 6802725 + 9523815 + 13333341 + 15873025 + 20408175 + 22222235 + 28571445 + 40000023 + 47619075 + 66666705 + 111111175 + 142857225 + 200000115 + 333333525 = 1083561009

Visual Basic .NET

Translation of: ALGOL 68

<lang vbnet>Module AbundantOddNumbers

   ' find some abundant odd numbers - numbers where the sum of the proper
   '                                  divisors is bigger than the number
   '                                  itself

   ' returns the sum of the proper divisors of n
   Private Function divisorSum(n As Integer) As Integer
       Dim sum As Integer = 1
       For d As Integer = 2 To Math.Round(Math.Sqrt(n))
           If n Mod d = 0 Then
               sum += d
               Dim otherD As Integer = n \ d
               IF otherD <> d Then
                   sum += otherD
               End If
           End If
       Next d
       Return sum
   End Function

   ' find numbers required by the task
   Public Sub Main(args() As String)
       ' first 25 odd abundant numbers
       Dim oddNumber As Integer = 1
       Dim aCount As Integer = 0
       Dim dSum As Integer = 0
       Console.Out.WriteLine("The first 25 abundant odd numbers:")
       Do While aCount < 25
           dSum = divisorSum(oddNumber)
           If dSum > oddNumber Then
               aCount += 1
               Console.Out.WriteLine(oddNumber.ToString.PadLeft(6) & " proper divisor sum: " & dSum)
           End If
           oddNumber += 2
       Loop
       ' 1000th odd abundant number
       Do While aCount < 1000
           dSum = divisorSum(oddNumber)
           If dSum > oddNumber Then
               aCount += 1
           End If
           oddNumber += 2
       Loop
       Console.Out.WriteLine("1000th abundant odd number:")
       Console.Out.WriteLine("    " & (oddNumber - 2) & " proper divisor sum: " & dSum)
       ' first odd abundant number > one billion
       oddNumber = 1000000001
       Dim found As Boolean = False
       Do While Not found
           dSum = divisorSum(oddNumber)
           If dSum > oddNumber Then
               found = True
               Console.Out.WriteLine("First abundant odd number > 1 000 000 000:")
               Console.Out.WriteLine("    " & oddNumber & " proper divisor sum: " & dSum)
           End If
           oddNumber += 2
       Loop
   End Sub

End Module</lang>

Output:

The first 25 abundant odd numbers:
   945 proper divisor sum: 975
  1575 proper divisor sum: 1649
  2205 proper divisor sum: 2241
  2835 proper divisor sum: 2973
  3465 proper divisor sum: 4023
  4095 proper divisor sum: 4641
  4725 proper divisor sum: 5195
  5355 proper divisor sum: 5877
  5775 proper divisor sum: 6129
  5985 proper divisor sum: 6495
  6435 proper divisor sum: 6669
  6615 proper divisor sum: 7065
  6825 proper divisor sum: 7063
  7245 proper divisor sum: 7731
  7425 proper divisor sum: 7455
  7875 proper divisor sum: 8349
  8085 proper divisor sum: 8331
  8415 proper divisor sum: 8433
  8505 proper divisor sum: 8967
  8925 proper divisor sum: 8931
  9135 proper divisor sum: 9585
  9555 proper divisor sum: 9597
  9765 proper divisor sum: 10203
 10395 proper divisor sum: 12645
 11025 proper divisor sum: 11946
1000th abundant odd number:
    492975 proper divisor sum: 519361
First abundant odd number > 1 000 000 000:
    1000000575 proper divisor sum: 1083561009

zkl

<lang zkl>fcn oddAbundants(startAt=3){ //--> iterator

  Walker.zero().tweak(fcn(rn){
     n:=rn.value;
     while(True){

sum:=0; foreach d in ([3.. n.toFloat().sqrt().toInt(), 2]){ if( (y:=n/d) *d != n) continue; sum += ((y==d) and y or y+d) } if(sum>n){ rn.set(n+2); return(n) } n+=2;

     }
  }.fp(Ref(startAt.isOdd and startAt or startAt+1)))

}</lang> <lang zkl>fcn oddDivisors(n){ // -->sorted List

  [3.. n.toFloat().sqrt().toInt(), 2].pump(List(1),'wrap(d){
     if( (y:=n/d) *d != n) return(Void.Skip);
     if (y==d) y else T(y,d)
   }).flatten().sort()

} fcn printOAs(oas){ // List | int

  foreach n in (vm.arglist.flatten()){ 
     ds:=oddDivisors(n);
     println("%6,d: %6,d = %s".fmt(n, ds.sum(0), ds.sort().concat(" + ")))
  }

}</lang> <lang zkl>oaw:=oddAbundants();

println("First 25 abundant odd numbers:"); oaw.walk(25) : printOAs(_);

println("\nThe one thousandth abundant odd number is:"); oaw.drop(1_000 - 25).value : printOAs(_);

println("\nThe first abundant odd number above one billion is:"); printOAs(oddAbundants(1_000_000_000).next());</lang>

Output:

   945:    975 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 105 + 135 + 189 + 315
 1,575:  1,649 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 315 + 525
 2,205:  2,241 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 49 + 63 + 105 + 147 + 245 + 315 + 441 + 735
 2,835:  2,973 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 315 + 405 + 567 + 945
 3,465:  4,023 = 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 165 + 231 + 315 + 385 + 495 + 693 + 1155
 4,095:  4,641 = 1 + 3 + 5 + 7 + 9 + 13 + 15 + 21 + 35 + 39 + 45 + 63 + 65 + 91 + 105 + 117 + 195 + 273 + 315 + 455 + 585 + 819 + 1365
 4,725:  5,195 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 27 + 35 + 45 + 63 + 75 + 105 + 135 + 175 + 189 + 225 + 315 + 525 + 675 + 945 + 1575
 5,355:  5,877 = 1 + 3 + 5 + 7 + 9 + 15 + 17 + 21 + 35 + 45 + 51 + 63 + 85 + 105 + 119 + 153 + 255 + 315 + 357 + 595 + 765 + 1071 + 1785
 5,775:  6,129 = 1 + 3 + 5 + 7 + 11 + 15 + 21 + 25 + 33 + 35 + 55 + 75 + 77 + 105 + 165 + 175 + 231 + 275 + 385 + 525 + 825 + 1155 + 1925
 5,985:  6,495 = 1 + 3 + 5 + 7 + 9 + 15 + 19 + 21 + 35 + 45 + 57 + 63 + 95 + 105 + 133 + 171 + 285 + 315 + 399 + 665 + 855 + 1197 + 1995
 6,435:  6,669 = 1 + 3 + 5 + 9 + 11 + 13 + 15 + 33 + 39 + 45 + 55 + 65 + 99 + 117 + 143 + 165 + 195 + 429 + 495 + 585 + 715 + 1287 + 2145
 6,615:  7,065 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 49 + 63 + 105 + 135 + 147 + 189 + 245 + 315 + 441 + 735 + 945 + 1323 + 2205
 6,825:  7,063 = 1 + 3 + 5 + 7 + 13 + 15 + 21 + 25 + 35 + 39 + 65 + 75 + 91 + 105 + 175 + 195 + 273 + 325 + 455 + 525 + 975 + 1365 + 2275
 7,245:  7,731 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 23 + 35 + 45 + 63 + 69 + 105 + 115 + 161 + 207 + 315 + 345 + 483 + 805 + 1035 + 1449 + 2415
 7,425:  7,455 = 1 + 3 + 5 + 9 + 11 + 15 + 25 + 27 + 33 + 45 + 55 + 75 + 99 + 135 + 165 + 225 + 275 + 297 + 495 + 675 + 825 + 1485 + 2475
 7,875:  8,349 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 125 + 175 + 225 + 315 + 375 + 525 + 875 + 1125 + 1575 + 2625
 8,085:  8,331 = 1 + 3 + 5 + 7 + 11 + 15 + 21 + 33 + 35 + 49 + 55 + 77 + 105 + 147 + 165 + 231 + 245 + 385 + 539 + 735 + 1155 + 1617 + 2695
 8,415:  8,433 = 1 + 3 + 5 + 9 + 11 + 15 + 17 + 33 + 45 + 51 + 55 + 85 + 99 + 153 + 165 + 187 + 255 + 495 + 561 + 765 + 935 + 1683 + 2805
 8,505:  8,967 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 243 + 315 + 405 + 567 + 945 + 1215 + 1701 + 2835
 8,925:  8,931 = 1 + 3 + 5 + 7 + 15 + 17 + 21 + 25 + 35 + 51 + 75 + 85 + 105 + 119 + 175 + 255 + 357 + 425 + 525 + 595 + 1275 + 1785 + 2975
 9,135:  9,585 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 29 + 35 + 45 + 63 + 87 + 105 + 145 + 203 + 261 + 315 + 435 + 609 + 1015 + 1305 + 1827 + 3045
 9,555:  9,597 = 1 + 3 + 5 + 7 + 13 + 15 + 21 + 35 + 39 + 49 + 65 + 91 + 105 + 147 + 195 + 245 + 273 + 455 + 637 + 735 + 1365 + 1911 + 3185
 9,765: 10,203 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 31 + 35 + 45 + 63 + 93 + 105 + 155 + 217 + 279 + 315 + 465 + 651 + 1085 + 1395 + 1953 + 3255
10,395: 12,645 = 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 27 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 135 + 165 + 189 + 231 + 297 + 315 + 385 + 495 + 693 + 945 + 1155 + 1485 + 2079 + 3465
11,025: 11,946 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675

The one thousandth abundant odd number is:
492,975: 519,361 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 313 + 315 + 525 + 939 + 1565 + 1575 + 2191 + 2817 + 4695 + 6573 + 7825 + 10955 + 14085 + 19719 + 23475 + 32865 + 54775 + 70425 + 98595 + 164325

The first abundant odd number above one billion is:
1,000,000,575: 1,083,561,009 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 + 11025 + 90703 + 272109 + 453515 + 634921 + 816327 + 1360545 + 1904763 + 2267575 + 3174605 + 4081635 + 4444447 + 5714289 + 6802725 + 9523815 + 13333341 + 15873025 + 20408175 + 22222235 + 28571445 + 40000023 + 47619075 + 66666705 + 111111175 + 142857225 + 200000115 + 333333525