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A peculiar triangular number (Posted on 2015-12-10) Difficulty: 3 of 5
28 is the smallest triangular number to have over five divisors (1,2,4,7,14,28).

What is the value of the smallest triangular number to have
over one hundred divisors? ... over two hundred divisors?

Based on a problem from Project Euler.

No Solution Yet Submitted by Ady TZIDON    
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Solution computer solution (spoiler) | Comment 1 of 2
The triangular numbers through the 3000th triangular number were checked. Any that had more than 100 factors was reported.

The first line lists the ordinality of the triangular number, the triangular number itself, and its number of factors. The prime factorization of each appears on the second line.

The first one to have more than 100 factors was the 384th triangular number, 73,920, which has 112 factors. 

The first one to have more than 200 factors was the 2015th triangular number, 2,031,120, which has 240 factors.


384 73920 112
2^6 * 3 * 5 * 7 * 11

440 97020 108
2^2 * 3^2 * 5 * 7^2 * 11

560 157080 128
2^3 * 3 * 5 * 7 * 11 * 17

575 165600 108
2^5 * 3^2 * 5^2 * 23

735 270480 120
2^4 * 3 * 5 * 7^2 * 23

896 401856 112
2^6 * 3 * 7 * 13 * 23

935 437580 144
2^2 * 3^2 * 5 * 11 * 13 * 17

944 446040 128
2^3 * 3^3 * 5 * 7 * 59

1088 592416 108
2^5 * 3^2 * 11^2 * 17

1104 609960 128
2^3 * 3 * 5 * 13 * 17 * 23

1175 690900 108
2^2 * 3 * 5^2 * 7^2 * 47

1215 738720 144
2^5 * 3^5 * 5 * 19

1224 749700 162
2^2 * 3^2 * 5^2 * 7^2 * 17

1280 819840 128
2^7 * 3 * 5 * 7 * 61

1287 828828 144
2^2 * 3^2 * 7 * 11 * 13 * 23

1295 839160 160
2^3 * 3^4 * 5 * 7 * 37

1364 930930 128
2 * 3 * 5 * 7 * 11 * 13 * 31

1407 990528 112
2^6 * 3 * 7 * 11 * 67

1440 1037520 120
2^4 * 3^2 * 5 * 11 * 131

1455 1059240 128
2^3 * 3 * 5 * 7 * 13 * 97

1484 1101870 128
2 * 3^3 * 5 * 7 * 11 * 53

1520 1155960 144
2^3 * 3^2 * 5 * 13^2 * 19

1539 1185030 160
2 * 3^4 * 5 * 7 * 11 * 19

1575 1241100 108
2^2 * 3^2 * 5^2 * 7 * 197

1595 1272810 128
2 * 3 * 5 * 7 * 11 * 19 * 29

1599 1279200 144
2^5 * 3 * 5^2 * 13 * 41

1664 1385280 168
2^6 * 3^2 * 5 * 13 * 37

1679 1410360 128
2^3 * 3 * 5 * 7 * 23 * 73

1700 1445850 144
2 * 3^5 * 5^2 * 7 * 17

1715 1471470 128
2 * 3 * 5 * 7^3 * 11 * 13

1728 1493856 192
2^5 * 3^3 * 7 * 13 * 19

1799 1619100 108
2^2 * 3^2 * 5^2 * 7 * 257

1824 1664400 120
2^4 * 3 * 5^2 * 19 * 73

1880 1768140 144
2^2 * 3^2 * 5 * 11 * 19 * 47

1904 1813560 128
2^3 * 3 * 5 * 7 * 17 * 127

1919 1842240 112
2^6 * 3 * 5 * 19 * 101

1920 1844160 112
2^6 * 3 * 5 * 17 * 113

1935 1873080 144
2^3 * 3^2 * 5 * 11^2 * 43

1952 1906128 120
2^4 * 3^2 * 7 * 31 * 61

2000 2001000 128
2^3 * 3 * 5^3 * 23 * 29

2015 2031120 240
2^4 * 3^2 * 5 * 7 * 13 * 31

2024 2049300 180
2^2 * 3^4 * 5^2 * 11 * 23

2064 2131080 128
2^3 * 3 * 5 * 7 * 43 * 59

2079 2162160 320
2^4 * 3^3 * 5 * 7 * 11 * 13

2144 2299440 160
2^4 * 3 * 5 * 11 * 13 * 67

2159 2331720 128
2^3 * 3^3 * 5 * 17 * 127

2175 2366400 168
2^6 * 3 * 5^2 * 17 * 29

2184 2386020 192
2^2 * 3 * 5 * 7 * 13 * 19 * 23

2204 2429910 144
2 * 3^2 * 5 * 7^2 * 19 * 29

2232 2492028 144
2^2 * 3^2 * 7 * 11 * 29 * 31

2240 2509920 192
2^5 * 3^3 * 5 * 7 * 83

2255 2543640 128
2^3 * 3 * 5 * 11 * 41 * 47

2288 2618616 128
2^3 * 3 * 7 * 11 * 13 * 109

2295 2634660 192
2^2 * 3^3 * 5 * 7 * 17 * 41

2303 2653056 144
2^7 * 3^2 * 7^2 * 47

2375 2821500 192
2^2 * 3^3 * 5^3 * 11 * 19

2379 2831010 128
2 * 3 * 5 * 7 * 13 * 17 * 61

2400 2881200 150
2^4 * 3 * 5^2 * 7^4

2415 2917320 128
2^3 * 3 * 5 * 7 * 23 * 151

2431 2956096 112
2^6 * 11 * 13 * 17 * 19

2464 3036880 160
2^4 * 5 * 7 * 11 * 17 * 29

2484 3086370 128
2 * 3^3 * 5 * 7 * 23 * 71

2499 3123750 120
2 * 3 * 5^4 * 7^2 * 17

2519 3173940 144
2^2 * 3^2 * 5 * 7 * 11 * 229

2583 3337236 144
2^2 * 3^2 * 7 * 17 * 19 * 41

2600 3381300 162
2^2 * 3^2 * 5^2 * 13 * 17^2

2624 3444000 192
2^5 * 3 * 5^3 * 7 * 41

2639 3483480 256
2^3 * 3 * 5 * 7 * 11 * 13 * 29

2640 3486120 128
2^3 * 3 * 5 * 11 * 19 * 139

2655 3525840 120
2^4 * 3^2 * 5 * 59 * 83

2664 3549780 144
2^2 * 3^2 * 5 * 13 * 37 * 41

2736 3744216 192
2^3 * 3^2 * 7 * 17 * 19 * 23

2744 3766140 144
2^2 * 3^2 * 5 * 7^3 * 61

2783 3873936 120
2^4 * 3 * 11^2 * 23 * 29

2799 3918600 144
2^3 * 3^2 * 5^2 * 7 * 311

2820 3977610 128
2 * 3 * 5 * 7 * 13 * 31 * 47

2880 4148640 144
2^5 * 3^2 * 5 * 43 * 67

2904 4218060 144
2^2 * 3 * 5 * 7 * 11^2 * 83

2915 4250070 112
2 * 3^6 * 5 * 11 * 53

2924 4276350 144
2 * 3^2 * 5^2 * 13 * 17 * 43

2925 4279275 144
3^2 * 5^2 * 7 * 11 * 13 * 19

2943 4332096 112
2^6 * 3^3 * 23 * 109

2944 4335040 112
2^6 * 5 * 19 * 23 * 31

2960 4382280 192
2^3 * 3^2 * 5 * 7 * 37 * 47

2975 4426800 240
2^4 * 3 * 5^2 * 7 * 17 * 31


from

DefDbl A-Z
Dim crlf$, fct(20, 1)


Private Sub Form_Load()
 Form1.Visible = True
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)
 
 tr = 1
 For addend = 2 To 3000
   DoEvents
   tr = tr + addend
   f = factor(tr)
   dvsrs = 1
   For i = 1 To f
     dvsrs = dvsrs * (fct(i, 1) + 1)
   Next
   If dvsrs > 100 Then
   Text1.Text = Text1.Text & addend & Str(tr) & Str(dvsrs) & crlf
     For i = 1 To f
       If i > 1 Then Text1.Text = Text1.Text & " * "
       Text1.Text = Text1.Text & fct(i, 0)
       If fct(i, 1) > 1 Then Text1.Text = Text1.Text & "^" & fct(i, 1)
     Next
   Text1.Text = Text1.Text & crlf & crlf
   End If
 Next


 Text1.Text = Text1.Text & crlf & " done"
  
End Sub

Function factor(num)
 diffCt = 0: good = 1
 n = Abs(num): If n > 0 Then limit = Sqr(n) Else limit = 0
 If limit <> Int(limit) Then limit = Int(limit + 1)
 dv = 2: GoSub DivideIt
 dv = 3: GoSub DivideIt
 dv = 5: GoSub DivideIt
 dv = 7
 Do Until dv > limit
   GoSub DivideIt: dv = dv + 4 '11
   GoSub DivideIt: dv = dv + 2 '13
   GoSub DivideIt: dv = dv + 4 '17
   GoSub DivideIt: dv = dv + 2 '19
   GoSub DivideIt: dv = dv + 4 '23
   GoSub DivideIt: dv = dv + 6 '29
   GoSub DivideIt: dv = dv + 2 '31
   GoSub DivideIt: dv = dv + 6 '37
   If INKEY$ = Chr$(27) Then s$ = Chr$(27): Exit Function
 Loop
 If n > 1 Then diffCt = diffCt + 1: fct(diffCt, 0) = n: fct(diffCt, 1) = 1
 factor = diffCt
 Exit Function

DivideIt:
 cnt = 0
 Do
  q = Int(n / dv)
  If q * dv = n And n > 0 Then
    n = q: cnt = cnt + 1: If n > 0 Then limit = Sqr(n) Else limit = 0
    If limit <> Int(limit) Then limit = Int(limit + 1)
   Else
    Exit Do
  End If
 Loop
 If cnt > 0 Then
   diffCt = diffCt + 1
   fct(diffCt, 0) = dv
   fct(diffCt, 1) = cnt
 End If
 Return
End Function


  Posted by Charlie on 2015-12-10 18:00:00
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