Abundant odd numbers

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Abundant odd numbers is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

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 / 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.


Task name and requirements requirements have changed, (and may change again) see discussion page for details.

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 main() { 

   const max = 25
   const maxOdd = 5
   fmt.Println("The first", max, "nice numbers are:")
   count := 0
   for n := 1; count < max; n++ {
       divs := divisors(n)
       if tot := sum(divs); tot > n {
           count++
           s := sumStr(divs)
           fmt.Printf("%2d. %3d => %s = %d > %d\n", count, n, s, tot, n)
       }
   }
   fmt.Println("\nThe first", maxOdd, "odd nice numbers are:")
   count = 0
   for n := 1; count < maxOdd; n += 2 {
       divs := divisors(n)
       if tot := sum(divs); tot > n {
           count++
           s := sumStr(divs)
           fmt.Printf("%d. %4d => %s = %d > %d\n", count, n, s, tot, n)
       }
   }

}</lang>

Output:
The first 25 nice numbers are:
 1.  12 => 1 + 2 + 3 + 4 + 6 = 16 > 12
 2.  18 => 1 + 2 + 3 + 6 + 9 = 21 > 18
 3.  20 => 1 + 2 + 4 + 5 + 10 = 22 > 20
 4.  24 => 1 + 2 + 3 + 4 + 6 + 8 + 12 = 36 > 24
 5.  30 => 1 + 2 + 3 + 5 + 6 + 10 + 15 = 42 > 30
 6.  36 => 1 + 2 + 3 + 4 + 6 + 9 + 12 + 18 = 55 > 36
 7.  40 => 1 + 2 + 4 + 5 + 8 + 10 + 20 = 50 > 40
 8.  42 => 1 + 2 + 3 + 6 + 7 + 14 + 21 = 54 > 42
 9.  48 => 1 + 2 + 3 + 4 + 6 + 8 + 12 + 16 + 24 = 76 > 48
10.  54 => 1 + 2 + 3 + 6 + 9 + 18 + 27 = 66 > 54
11.  56 => 1 + 2 + 4 + 7 + 8 + 14 + 28 = 64 > 56
12.  60 => 1 + 2 + 3 + 4 + 5 + 6 + 10 + 12 + 15 + 20 + 30 = 108 > 60
13.  66 => 1 + 2 + 3 + 6 + 11 + 22 + 33 = 78 > 66
14.  70 => 1 + 2 + 5 + 7 + 10 + 14 + 35 = 74 > 70
15.  72 => 1 + 2 + 3 + 4 + 6 + 8 + 9 + 12 + 18 + 24 + 36 = 123 > 72
16.  78 => 1 + 2 + 3 + 6 + 13 + 26 + 39 = 90 > 78
17.  80 => 1 + 2 + 4 + 5 + 8 + 10 + 16 + 20 + 40 = 106 > 80
18.  84 => 1 + 2 + 3 + 4 + 6 + 7 + 12 + 14 + 21 + 28 + 42 = 140 > 84
19.  88 => 1 + 2 + 4 + 8 + 11 + 22 + 44 = 92 > 88
20.  90 => 1 + 2 + 3 + 5 + 6 + 9 + 10 + 15 + 18 + 30 + 45 = 144 > 90
21.  96 => 1 + 2 + 3 + 4 + 6 + 8 + 12 + 16 + 24 + 32 + 48 = 156 > 96
22. 100 => 1 + 2 + 4 + 5 + 10 + 20 + 25 + 50 = 117 > 100
23. 102 => 1 + 2 + 3 + 6 + 17 + 34 + 51 = 114 > 102
24. 104 => 1 + 2 + 4 + 8 + 13 + 26 + 52 = 106 > 104
25. 108 => 1 + 2 + 3 + 4 + 6 + 9 + 12 + 18 + 27 + 36 + 54 = 172 > 108

The first 5 odd nice numbers are:
1.  945 => 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 105 + 135 + 189 + 315 = 975 > 945
2. 1575 => 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 315 + 525 = 1649 > 1575
3. 2205 => 1 + 3 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 49 + 63 + 105 + 147 + 245 + 315 + 441 + 735 = 2241 > 2205
4. 2835 => 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 315 + 405 + 567 + 945 = 2973 > 2835
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 > 3465

Perl 6

Works with: Rakudo version 2019.03

Takes a little over 5 seconds to complete (on my not-very-fast system).

<lang perl6>sub abundant (\x) { 

   my @l = x.is-prime ?? 1 !! flat
   1, (2 .. x.sqrt.floor).map: -> \d {
        my \y = x div d;
        next if y * d !== x;
        d !== y ?? (d, y) !! d
   };
   (my $s = @l.sum) > x ?? (@l.sort) !! Empty;

}

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

   $start-at = ( $start-at + 14 ) div 15;
   $start-at += $start-at %% 2;
   $start-at *= 15;
   ($start-at, *+30 ... *).hyper.map: {
       next unless my $oa = .&abundant;
       sprintf "%6d: divisor sum: {$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:
498225: divisor sum: 1 + 3 + 5 + 7 + 13 + 15 + 21 + 25 + 35 + 39 + 65 + 73 + 75 + 91 + 105 + 175 + 195 + 219 + 273 + 325 + 365 + 455 + 511 + 525 + 949 + 975 + 1095 + 1365 + 1533 + 1825 + 2275 + 2555 + 2847 + 4745 + 5475 + 6643 + 6825 + 7665 + 12775 + 14235 + 19929 + 23725 + 33215 + 38325 + 71175 + 99645 + 166075 = 529487

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

REXX

<lang rexx>/*REXX pgm displays N nice numbers, where nice #'s are whose sum of its factors > #. */ parse arg N . /*obtain optional arguments from the CL*/ if N== | N=="," then N= 25 /*Not specified? Then use the default.*/ w= length(N); ww= w+ w /*used for aligning the columnar output*/

  1. = 0 /*count of nice numbers (so far). */
     do j=1  until #>=N;     $=sigma(j)         /*get sigma for an integer.            */
     if $<=j  then iterate                      /*sigma  +  J ?    Then ignore it.     */
     #= # + 1
     say 'nice number '   right(#,w)   " is:"   right(j,ww)',     sigma is:'  right($,ww)
     end  /*j*/

exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ sigma: procedure; parse arg x; if x<2 then return 0; odd= x // 2 /* // ◄──remainder.*/ 

      s= 1                                      /* [↓]  only use  EVEN or ODD integers.*/
            do j=2+odd  by 1+odd  while j*j<x   /*divide by all integers up to  √x.    */
            if x//j==0  then  s= s + j +  x%j   /*add the two divisors to (sigma) sum. */
            end   /*j*/                         /* [↑]  %  is the REXX integer division*/
                                                /* [↓]  adjust for a square.       ___ */
      if j*j==x  then  return s + j             /*Was  X  a square?   If so, add  √ x  */
                       return s                 /*return (sigma) sum of the divisors.  */</lang>
output   when using the default input:
nice number   1  is:   12,     sigma is:   16
nice number   2  is:   18,     sigma is:   21
nice number   3  is:   20,     sigma is:   22
nice number   4  is:   24,     sigma is:   36
nice number   5  is:   30,     sigma is:   42
nice number   6  is:   36,     sigma is:   55
nice number   7  is:   40,     sigma is:   50
nice number   8  is:   42,     sigma is:   54
nice number   9  is:   48,     sigma is:   76
nice number  10  is:   54,     sigma is:   66
nice number  11  is:   56,     sigma is:   64
nice number  12  is:   60,     sigma is:  108
nice number  13  is:   66,     sigma is:   78
nice number  14  is:   70,     sigma is:   74
nice number  15  is:   72,     sigma is:  123
nice number  16  is:   78,     sigma is:   90
nice number  17  is:   80,     sigma is:  106
nice number  18  is:   84,     sigma is:  140
nice number  19  is:   88,     sigma is:   92
nice number  20  is:   90,     sigma is:  144
nice number  21  is:   96,     sigma is:  156
nice number  22  is:  100,     sigma is:  117
nice number  23  is:  102,     sigma is:  114
nice number  24  is:  104,     sigma is:  106
nice number  25  is:  108,     sigma is:  172

Ring

<lang ring>

  1. Project: Nice numbers

max = 25 nr = 0 m = 1 check = 0 index = 0 while true 

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

end

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
       showArray(n,nArray,sum,index)
    ok

func showArray(n,nArray,sum,index) 

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

</lang>

Output:
1. 12 => 1 + 2 + 3 + 4 + 6 = 16 > 12
2. 18 => 1 + 2 + 3 + 6 + 9 = 21 > 18
3. 20 => 1 + 2 + 4 + 5 + 10 = 22 > 20
4. 24 => 1 + 2 + 3 + 4 + 6 + 8 + 12 = 36 > 24
5. 30 => 1 + 2 + 3 + 5 + 6 + 10 + 15 = 42 > 30
6. 36 => 1 + 2 + 3 + 4 + 6 + 9 + 12 + 18 = 55 > 36
7. 40 => 1 + 2 + 4 + 5 + 8 + 10 + 20 = 50 > 40
8. 42 => 1 + 2 + 3 + 6 + 7 + 14 + 21 = 54 > 42
9. 48 => 1 + 2 + 3 + 4 + 6 + 8 + 12 + 16 + 24 = 76 > 48
10. 54 => 1 + 2 + 3 + 6 + 9 + 18 + 27 = 66 > 54
11. 56 => 1 + 2 + 4 + 7 + 8 + 14 + 28 = 64 > 56
12. 60 => 1 + 2 + 3 + 4 + 5 + 6 + 10 + 12 + 15 + 20 + 30 = 108 > 60
13. 66 => 1 + 2 + 3 + 6 + 11 + 22 + 33 = 78 > 66
14. 70 => 1 + 2 + 5 + 7 + 10 + 14 + 35 = 74 > 70
15. 72 => 1 + 2 + 3 + 4 + 6 + 8 + 9 + 12 + 18 + 24 + 36 = 123 > 72
16. 78 => 1 + 2 + 3 + 6 + 13 + 26 + 39 = 90 > 78
17. 80 => 1 + 2 + 4 + 5 + 8 + 10 + 16 + 20 + 40 = 106 > 80
18. 84 => 1 + 2 + 3 + 4 + 6 + 7 + 12 + 14 + 21 + 28 + 42 = 140 > 84
19. 88 => 1 + 2 + 4 + 8 + 11 + 22 + 44 = 92 > 88
20. 90 => 1 + 2 + 3 + 5 + 6 + 9 + 10 + 15 + 18 + 30 + 45 = 144 > 90
21. 96 => 1 + 2 + 3 + 4 + 6 + 8 + 12 + 16 + 24 + 32 + 48 = 156 > 96
22. 100 => 1 + 2 + 4 + 5 + 10 + 20 + 25 + 50 = 117 > 100
23. 102 => 1 + 2 + 3 + 6 + 17 + 34 + 51 = 114 > 102
24. 104 => 1 + 2 + 4 + 8 + 13 + 26 + 52 = 106 > 104
25. 108 => 1 + 2 + 3 + 4 + 6 + 9 + 12 + 18 + 27 + 36 + 54 = 172 > 108
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