Highly Composite Numbers (Anti-Primes)
Thursday, December 6, 2018
By: Matthew Doucette

Highly composite numbers have more factors than any number smaller than themselves. In other words, they are numbers with a high degree of factorization or divisibility. This is the opposite of a prime number, which has no factors (other than 1 and itself).

 

Highly Composite Numbers (HCMs) and Factorizations:

1: 1.

2: 1, 2.

4: 1, 2, 4.

6: 1, 2, 3, 6.

12: 1, 2, 3, 4, 6, 12.

24: 1, 2, 3, 4, 6, 8, 12, 24.

36: 1, 2, 3, 4, 6, 9, 12, 18, 36.

48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.

60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.

120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120.

180: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180.

240: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240.

360: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360.

720: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 30, 36, 40, 45, 48, 60, 72, 80, 90, 120, 144, 180, 240, 360, 720.

840: 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 20, 21, 24, 28, 30, 35, 40, 42, 56, 60, 70, 84, 105, 120, 140, 168, 210, 280, 420, 840.

1260: 1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 14, 15, 18, 20, 21, 28, 30, 35, 36, 42, 45, 60, 63, 70, 84, 90, 105, 126, 140, 180, 210, 252, 315, 420, 630, 1260.

1680: 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 16, 20, 21, 24, 28, 30, 35, 40, 42, 48, 56, 60, 70, 80, 84, 105, 112, 120, 140, 168, 210, 240, 280, 336, 420, 560, 840, 1680.

2520: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42, 45, 56, 60, 63, 70, 72, 84, 90, 105, 120, 126, 140, 168, 180, 210, 252, 280, 315, 360, 420, 504, 630, 840, 1260, 2520.

5040: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42, 45, 48, 56, 60, 63, 70, 72, 80, 84, 90, 105, 112, 120, 126, 140, 144, 168, 180, 210, 240, 252, 280, 315, 336, 360, 420, 504, 560, 630, 720, 840, 1008, 1260, 1680, 2520, 5040.

7560: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 21, 24, 27, 28, 30, 35, 36, 40, 42, 45, 54, 56, 60, 63, 70, 72, 84, 90, 105, 108, 120, 126, 135, 140, 168, 180, 189, 210, 216, 252, 270, 280, 315, 360, 378, 420, 504, 540, 630, 756, 840, 945, 1080, 1260, 1512, 1890, 2520, 3780, 7560.

10080: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 32, 35, 36, 40, 42, 45, 48, 56, 60, 63, 70, 72, 80, 84, 90, 96, 105, 112, 120, 126, 140, 144, 160, 168, 180, 210, 224, 240, 252, 280, 288, 315, 336, 360, 420, 480, 504, 560, 630, 672, 720, 840, 1008, 1120, 1260, 1440, 1680, 2016, 2520, 3360, 5040, 10080.

15120: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 27, 28, 30, 35, 36, 40, 42, 45, 48, 54, 56, 60, 63, 70, 72, 80, 84, 90, 105, 108, 112, 120, 126, 135, 140, 144, 168, 180, 189, 210, 216, 240, 252, 270, 280, 315, 336, 360, 378, 420, 432, 504, 540, 560, 630, 720, 756, 840, 945, 1008, 1080, 1260, 1512, 1680, 1890, 2160, 2520, 3024, 3780, 5040, 7560, 15120.

20160: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 32, 35, 36, 40, 42, 45, 48, 56, 60, 63, 64, 70, 72, 80, 84, 90, 96, 105, 112, 120, 126, 140, 144, 160, 168, 180, 192, 210, 224, 240, 252, 280, 288, 315, 320, 336, 360, 420, 448, 480, 504, 560, 576, 630, 672, 720, 840, 960, 1008, 1120, 1260, 1344, 1440, 1680, 2016, 2240, 2520, 2880, 3360, 4032, 5040, 6720, 10080, 20160.

25200: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 28, 30, 35, 36, 40, 42, 45, 48, 50, 56, 60, 63, 70, 72, 75, 80, 84, 90, 100, 105, 112, 120, 126, 140, 144, 150, 168, 175, 180, 200, 210, 225, 240, 252, 280, 300, 315, 336, 350, 360, 400, 420, 450, 504, 525, 560, 600, 630, 700, 720, 840, 900, 1008, 1050, 1200, 1260, 1400, 1575, 1680, 1800, 2100, 2520, 2800, 3150, 3600, 4200, 5040, 6300, 8400, 12600, 25200.

27720: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 18, 20, 21, 22, 24, 28, 30, 33, 35, 36, 40, 42, 44, 45, 55, 56, 60, 63, 66, 70, 72, 77, 84, 88, 90, 99, 105, 110, 120, 126, 132, 140, 154, 165, 168, 180, 198, 210, 220, 231, 252, 264, 280, 308, 315, 330, 360, 385, 396, 420, 440, 462, 495, 504, 616, 630, 660, 693, 770, 792, 840, 924, 990, 1155, 1260, 1320, 1386, 1540, 1848, 1980, 2310, 2520, 2772, 3080, 3465, 3960, 4620, 5544, 6930, 9240, 13860, 27720.

45360: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 27, 28, 30, 35, 36, 40, 42, 45, 48, 54, 56, 60, 63, 70, 72, 80, 81, 84, 90, 105, 108, 112, 120, 126, 135, 140, 144, 162, 168, 180, 189, 210, 216, 240, 252, 270, 280, 315, 324, 336, 360, 378, 405, 420, 432, 504, 540, 560, 567, 630, 648, 720, 756, 810, 840, 945, 1008, 1080, 1134, 1260, 1296, 1512, 1620, 1680, 1890, 2160, 2268, 2520, 2835, 3024, 3240, 3780, 4536, 5040, 5670, 6480, 7560, 9072, 11340, 15120, 22680, 45360.

50400: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 28, 30, 32, 35, 36, 40, 42, 45, 48, 50, 56, 60, 63, 70, 72, 75, 80, 84, 90, 96, 100, 105, 112, 120, 126, 140, 144, 150, 160, 168, 175, 180, 200, 210, 224, 225, 240, 252, 280, 288, 300, 315, 336, 350, 360, 400, 420, 450, 480, 504, 525, 560, 600, 630, 672, 700, 720, 800, 840, 900, 1008, 1050, 1120, 1200, 1260, 1400, 1440, 1575, 1680, 1800, 2016, 2100, 2400, 2520, 2800, 3150, 3360, 3600, 4200, 5040, 5600, 6300, 7200, 8400, 10080, 12600, 16800, 25200, 50400.

55440: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 28, 30, 33, 35, 36, 40, 42, 44, 45, 48, 55, 56, 60, 63, 66, 70, 72, 77, 80, 84, 88, 90, 99, 105, 110, 112, 120, 126, 132, 140, 144, 154, 165, 168, 176, 180, 198, 210, 220, 231, 240, 252, 264, 280, 308, 315, 330, 336, 360, 385, 396, 420, 440, 462, 495, 504, 528, 560, 616, 630, 660, 693, 720, 770, 792, 840, 880, 924, 990, 1008, 1155, 1232, 1260, 1320, 1386, 1540, 1584, 1680, 1848, 1980, 2310, 2520, 2640, 2772, 3080, 3465, 3696, 3960, 4620, 5040, 5544, 6160, 6930, 7920, 9240, 11088, 13860, 18480, 27720, 55440.

83160: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 18, 20, 21, 22, 24, 27, 28, 30, 33, 35, 36, 40, 42, 44, 45, 54, 55, 56, 60, 63, 66, 70, 72, 77, 84, 88, 90, 99, 105, 108, 110, 120, 126, 132, 135, 140, 154, 165, 168, 180, 189, 198, 210, 216, 220, 231, 252, 264, 270, 280, 297, 308, 315, 330, 360, 378, 385, 396, 420, 440, 462, 495, 504, 540, 594, 616, 630, 660, 693, 756, 770, 792, 840, 924, 945, 990, 1080, 1155, 1188, 1260, 1320, 1386, 1485, 1512, 1540, 1848, 1890, 1980, 2079, 2310, 2376, 2520, 2772, 2970, 3080, 3465, 3780, 3960, 4158, 4620, 5544, 5940, 6930, 7560, 8316, 9240, 10395, 11880, 13860, 16632, 20790, 27720, 41580, 83160.

110880: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 28, 30, 32, 33, 35, 36, 40, 42, 44, 45, 48, 55, 56, 60, 63, 66, 70, 72, 77, 80, 84, 88, 90, 96, 99, 105, 110, 112, 120, 126, 132, 140, 144, 154, 160, 165, 168, 176, 180, 198, 210, 220, 224, 231, 240, 252, 264, 280, 288, 308, 315, 330, 336, 352, 360, 385, 396, 420, 440, 462, 480, 495, 504, 528, 560, 616, 630, 660, 672, 693, 720, 770, 792, 840, 880, 924, 990, 1008, 1056, 1120, 1155, 1232, 1260, 1320, 1386, 1440, 1540, 1584, 1680, 1760, 1848, 1980, 2016, 2310, 2464, 2520, 2640, 2772, 3080, 3168, 3360, 3465, 3696, 3960, 4620, 5040, 5280, 5544, 6160, 6930, 7392, 7920, 9240, 10080, 11088, 12320, 13860, 15840, 18480, 22176, 27720, 36960, 55440, 110880.

166320: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 27, 28, 30, 33, 35, 36, 40, 42, 44, 45, 48, 54, 55, 56, 60, 63, 66, 70, 72, 77, 80, 84, 88, 90, 99, 105, 108, 110, 112, 120, 126, 132, 135, 140, 144, 154, 165, 168, 176, 180, 189, 198, 210, 216, 220, 231, 240, 252, 264, 270, 280, 297, 308, 315, 330, 336, 360, 378, 385, 396, 420, 432, 440, 462, 495, 504, 528, 540, 560, 594, 616, 630, 660, 693, 720, 756, 770, 792, 840, 880, 924, 945, 990, 1008, 1080, 1155, 1188, 1232, 1260, 1320, 1386, 1485, 1512, 1540, 1584, 1680, 1848, 1890, 1980, 2079, 2160, 2310, 2376, 2520, 2640, 2772, 2970, 3024, 3080, 3465, 3696, 3780, 3960, 4158, 4620, 4752, 5040, 5544, 5940, 6160, 6930, 7560, 7920, 8316, 9240, 10395, 11088, 11880, 13860, 15120, 16632, 18480, 20790, 23760, 27720, 33264, 41580, 55440, 83160, 166320.

221760: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 28, 30, 32, 33, 35, 36, 40, 42, 44, 45, 48, 55, 56, 60, 63, 64, 66, 70, 72, 77, 80, 84, 88, 90, 96, 99, 105, 110, 112, 120, 126, 132, 140, 144, 154, 160, 165, 168, 176, 180, 192, 198, 210, 220, 224, 231, 240, 252, 264, 280, 288, 308, 315, 320, 330, 336, 352, 360, 385, 396, 420, 440, 448, 462, 480, 495, 504, 528, 560, 576, 616, 630, 660, 672, 693, 704, 720, 770, 792, 840, 880, 924, 960, 990, 1008, 1056, 1120, 1155, 1232, 1260, 1320, 1344, 1386, 1440, 1540, 1584, 1680, 1760, 1848, 1980, 2016, 2112, 2240, 2310, 2464, 2520, 2640, 2772, 2880, 3080, 3168, 3360, 3465, 3520, 3696, 3960, 4032, 4620, 4928, 5040, 5280, 5544, 6160, 6336, 6720, 6930, 7392, 7920, 9240, 10080, 10560, 11088, 12320, 13860, 14784, 15840, 18480, 20160, 22176, 24640, 27720, 31680, 36960, 44352, 55440, 73920, 110880, 221760.

277200: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 28, 30, 33, 35, 36, 40, 42, 44, 45, 48, 50, 55, 56, 60, 63, 66, 70, 72, 75, 77, 80, 84, 88, 90, 99, 100, 105, 110, 112, 120, 126, 132, 140, 144, 150, 154, 165, 168, 175, 176, 180, 198, 200, 210, 220, 225, 231, 240, 252, 264, 275, 280, 300, 308, 315, 330, 336, 350, 360, 385, 396, 400, 420, 440, 450, 462, 495, 504, 525, 528, 550, 560, 600, 616, 630, 660, 693, 700, 720, 770, 792, 825, 840, 880, 900, 924, 990, 1008, 1050, 1100, 1155, 1200, 1232, 1260, 1320, 1386, 1400, 1540, 1575, 1584, 1650, 1680, 1800, 1848, 1925, 1980, 2100, 2200, 2310, 2475, 2520, 2640, 2772, 2800, 3080, 3150, 3300, 3465, 3600, 3696, 3850, 3960, 4200, 4400, 4620, 4950, 5040, 5544, 5775, 6160, 6300, 6600, 6930, 7700, 7920, 8400, 9240, 9900, 11088, 11550, 12600, 13200, 13860, 15400, 17325, 18480, 19800, 23100, 25200, 27720, 30800, 34650, 39600, 46200, 55440, 69300, 92400, 138600, 277200.

332640: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 27, 28, 30, 32, 33, 35, 36, 40, 42, 44, 45, 48, 54, 55, 56, 60, 63, 66, 70, 72, 77, 80, 84, 88, 90, 96, 99, 105, 108, 110, 112, 120, 126, 132, 135, 140, 144, 154, 160, 165, 168, 176, 180, 189, 198, 210, 216, 220, 224, 231, 240, 252, 264, 270, 280, 288, 297, 308, 315, 330, 336, 352, 360, 378, 385, 396, 420, 432, 440, 462, 480, 495, 504, 528, 540, 560, 594, 616, 630, 660, 672, 693, 720, 756, 770, 792, 840, 864, 880, 924, 945, 990, 1008, 1056, 1080, 1120, 1155, 1188, 1232, 1260, 1320, 1386, 1440, 1485, 1512, 1540, 1584, 1680, 1760, 1848, 1890, 1980, 2016, 2079, 2160, 2310, 2376, 2464, 2520, 2640, 2772, 2970, 3024, 3080, 3168, 3360, 3465, 3696, 3780, 3960, 4158, 4320, 4620, 4752, 5040, 5280, 5544, 5940, 6048, 6160, 6930, 7392, 7560, 7920, 8316, 9240, 9504, 10080, 10395, 11088, 11880, 12320, 13860, 15120, 15840, 16632, 18480, 20790, 22176, 23760, 27720, 30240, 33264, 36960, 41580, 47520, 55440, 66528, 83160, 110880, 166320, 332640.

498960: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 27, 28, 30, 33, 35, 36, 40, 42, 44, 45, 48, 54, 55, 56, 60, 63, 66, 70, 72, 77, 80, 81, 84, 88, 90, 99, 105, 108, 110, 112, 120, 126, 132, 135, 140, 144, 154, 162, 165, 168, 176, 180, 189, 198, 210, 216, 220, 231, 240, 252, 264, 270, 280, 297, 308, 315, 324, 330, 336, 360, 378, 385, 396, 405, 420, 432, 440, 462, 495, 504, 528, 540, 560, 567, 594, 616, 630, 648, 660, 693, 720, 756, 770, 792, 810, 840, 880, 891, 924, 945, 990, 1008, 1080, 1134, 1155, 1188, 1232, 1260, 1296, 1320, 1386, 1485, 1512, 1540, 1584, 1620, 1680, 1782, 1848, 1890, 1980, 2079, 2160, 2268, 2310, 2376, 2520, 2640, 2772, 2835, 2970, 3024, 3080, 3240, 3465, 3564, 3696, 3780, 3960, 4158, 4455, 4536, 4620, 4752, 5040, 5544, 5670, 5940, 6160, 6237, 6480, 6930, 7128, 7560, 7920, 8316, 8910, 9072, 9240, 10395, 11088, 11340, 11880, 12474, 13860, 14256, 15120, 16632, 17820, 18480, 20790, 22680, 23760, 24948, 27720, 31185, 33264, 35640, 41580, 45360, 49896, 55440, 62370, 71280, 83160, 99792, 124740, 166320, 249480, 498960.

554400: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 28, 30, 32, 33, 35, 36, 40, 42, 44, 45, 48, 50, 55, 56, 60, 63, 66, 70, 72, 75, 77, 80, 84, 88, 90, 96, 99, 100, 105, 110, 112, 120, 126, 132, 140, 144, 150, 154, 160, 165, 168, 175, 176, 180, 198, 200, 210, 220, 224, 225, 231, 240, 252, 264, 275, 280, 288, 300, 308, 315, 330, 336, 350, 352, 360, 385, 396, 400, 420, 440, 450, 462, 480, 495, 504, 525, 528, 550, 560, 600, 616, 630, 660, 672, 693, 700, 720, 770, 792, 800, 825, 840, 880, 900, 924, 990, 1008, 1050, 1056, 1100, 1120, 1155, 1200, 1232, 1260, 1320, 1386, 1400, 1440, 1540, 1575, 1584, 1650, 1680, 1760, 1800, 1848, 1925, 1980, 2016, 2100, 2200, 2310, 2400, 2464, 2475, 2520, 2640, 2772, 2800, 3080, 3150, 3168, 3300, 3360, 3465, 3600, 3696, 3850, 3960, 4200, 4400, 4620, 4950, 5040, 5280, 5544, 5600, 5775, 6160, 6300, 6600, 6930, 7200, 7392, 7700, 7920, 8400, 8800, 9240, 9900, 10080, 11088, 11550, 12320, 12600, 13200, 13860, 15400, 15840, 16800, 17325, 18480, 19800, 22176, 23100, 25200, 26400, 27720, 30800, 34650, 36960, 39600, 46200, 50400, 55440, 61600, 69300, 79200, 92400, 110880, 138600, 184800, 277200, 554400.

665280: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 27, 28, 30, 32, 33, 35, 36, 40, 42, 44, 45, 48, 54, 55, 56, 60, 63, 64, 66, 70, 72, 77, 80, 84, 88, 90, 96, 99, 105, 108, 110, 112, 120, 126, 132, 135, 140, 144, 154, 160, 165, 168, 176, 180, 189, 192, 198, 210, 216, 220, 224, 231, 240, 252, 264, 270, 280, 288, 297, 308, 315, 320, 330, 336, 352, 360, 378, 385, 396, 420, 432, 440, 448, 462, 480, 495, 504, 528, 540, 560, 576, 594, 616, 630, 660, 672, 693, 704, 720, 756, 770, 792, 840, 864, 880, 924, 945, 960, 990, 1008, 1056, 1080, 1120, 1155, 1188, 1232, 1260, 1320, 1344, 1386, 1440, 1485, 1512, 1540, 1584, 1680, 1728, 1760, 1848, 1890, 1980, 2016, 2079, 2112, 2160, 2240, 2310, 2376, 2464, 2520, 2640, 2772, 2880, 2970, 3024, 3080, 3168, 3360, 3465, 3520, 3696, 3780, 3960, 4032, 4158, 4320, 4620, 4752, 4928, 5040, 5280, 5544, 5940, 6048, 6160, 6336, 6720, 6930, 7392, 7560, 7920, 8316, 8640, 9240, 9504, 10080, 10395, 10560, 11088, 11880, 12096, 12320, 13860, 14784, 15120, 15840, 16632, 18480, 19008, 20160, 20790, 22176, 23760, 24640, 27720, 30240, 31680, 33264, 36960, 41580, 44352, 47520, 55440, 60480, 66528, 73920, 83160, 95040, 110880, 133056, 166320, 221760, 332640, 665280.

720720: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 26, 28, 30, 33, 35, 36, 39, 40, 42, 44, 45, 48, 52, 55, 56, 60, 63, 65, 66, 70, 72, 77, 78, 80, 84, 88, 90, 91, 99, 104, 105, 110, 112, 117, 120, 126, 130, 132, 140, 143, 144, 154, 156, 165, 168, 176, 180, 182, 195, 198, 208, 210, 220, 231, 234, 240, 252, 260, 264, 273, 280, 286, 308, 312, 315, 330, 336, 360, 364, 385, 390, 396, 420, 429, 440, 455, 462, 468, 495, 504, 520, 528, 546, 560, 572, 585, 616, 624, 630, 660, 693, 715, 720, 728, 770, 780, 792, 819, 840, 858, 880, 910, 924, 936, 990, 1001, 1008, 1040, 1092, 1144, 1155, 1170, 1232, 1260, 1287, 1320, 1365, 1386, 1430, 1456, 1540, 1560, 1584, 1638, 1680, 1716, 1820, 1848, 1872, 1980, 2002, 2145, 2184, 2288, 2310, 2340, 2520, 2574, 2640, 2730, 2772, 2860, 3003, 3080, 3120, 3276, 3432, 3465, 3640, 3696, 3960, 4004, 4095, 4290, 4368, 4620, 4680, 5005, 5040, 5148, 5460, 5544, 5720, 6006, 6160, 6435, 6552, 6864, 6930, 7280, 7920, 8008, 8190, 8580, 9009, 9240, 9360, 10010, 10296, 10920, 11088, 11440, 12012, 12870, 13104, 13860, 15015, 16016, 16380, 17160, 18018, 18480, 20020, 20592, 21840, 24024, 25740, 27720, 30030, 32760, 34320, 36036, 40040, 45045, 48048, 51480, 55440, 60060, 65520, 72072, 80080, 90090, 102960, 120120, 144144, 180180, 240240, 360360, 720720.

1081080: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 20, 21, 22, 24, 26, 27, 28, 30, 33, 35, 36, 39, 40, 42, 44, 45, 52, 54, 55, 56, 60, 63, 65, 66, 70, 72, 77, 78, 84, 88, 90, 91, 99, 104, 105, 108, 110, 117, 120, 126, 130, 132, 135, 140, 143, 154, 156, 165, 168, 180, 182, 189, 195, 198, 210, 216, 220, 231, 234, 252, 260, 264, 270, 273, 280, 286, 297, 308, 312, 315, 330, 351, 360, 364, 378, 385, 390, 396, 420, 429, 440, 455, 462, 468, 495, 504, 520, 540, 546, 572, 585, 594, 616, 630, 660, 693, 702, 715, 728, 756, 770, 780, 792, 819, 840, 858, 910, 924, 936, 945, 990, 1001, 1080, 1092, 1144, 1155, 1170, 1188, 1260, 1287, 1320, 1365, 1386, 1404, 1430, 1485, 1512, 1540, 1560, 1638, 1716, 1755, 1820, 1848, 1890, 1980, 2002, 2079, 2145, 2184, 2310, 2340, 2376, 2457, 2520, 2574, 2730, 2772, 2808, 2860, 2970, 3003, 3080, 3276, 3432, 3465, 3510, 3640, 3780, 3861, 3960, 4004, 4095, 4158, 4290, 4620, 4680, 4914, 5005, 5148, 5460, 5544, 5720, 5940, 6006, 6435, 6552, 6930, 7020, 7560, 7722, 8008, 8190, 8316, 8580, 9009, 9240, 9828, 10010, 10296, 10395, 10920, 11880, 12012, 12285, 12870, 13860, 14040, 15015, 15444, 16380, 16632, 17160, 18018, 19305, 19656, 20020, 20790, 24024, 24570, 25740, 27027, 27720, 30030, 30888, 32760, 36036, 38610, 40040, 41580, 45045, 49140, 51480, 54054, 60060, 72072, 77220, 83160, 90090, 98280, 108108, 120120, 135135, 154440, 180180, 216216, 270270, 360360, 540540, 1081080.

1441440: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 26, 28, 30, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 52, 55, 56, 60, 63, 65, 66, 70, 72, 77, 78, 80, 84, 88, 90, 91, 96, 99, 104, 105, 110, 112, 117, 120, 126, 130, 132, 140, 143, 144, 154, 156, 160, 165, 168, 176, 180, 182, 195, 198, 208, 210, 220, 224, 231, 234, 240, 252, 260, 264, 273, 280, 286, 288, 308, 312, 315, 330, 336, 352, 360, 364, 385, 390, 396, 416, 420, 429, 440, 455, 462, 468, 480, 495, 504, 520, 528, 546, 560, 572, 585, 616, 624, 630, 660, 672, 693, 715, 720, 728, 770, 780, 792, 819, 840, 858, 880, 910, 924, 936, 990, 1001, 1008, 1040, 1056, 1092, 1120, 1144, 1155, 1170, 1232, 1248, 1260, 1287, 1320, 1365, 1386, 1430, 1440, 1456, 1540, 1560, 1584, 1638, 1680, 1716, 1760, 1820, 1848, 1872, 1980, 2002, 2016, 2080, 2145, 2184, 2288, 2310, 2340, 2464, 2520, 2574, 2640, 2730, 2772, 2860, 2912, 3003, 3080, 3120, 3168, 3276, 3360, 3432, 3465, 3640, 3696, 3744, 3960, 4004, 4095, 4290, 4368, 4576, 4620, 4680, 5005, 5040, 5148, 5280, 5460, 5544, 5720, 6006, 6160, 6240, 6435, 6552, 6864, 6930, 7280, 7392, 7920, 8008, 8190, 8580, 8736, 9009, 9240, 9360, 10010, 10080, 10296, 10920, 11088, 11440, 12012, 12320, 12870, 13104, 13728, 13860, 14560, 15015, 15840, 16016, 16380, 17160, 18018, 18480, 18720, 20020, 20592, 21840, 22176, 22880, 24024, 25740, 26208, 27720, 30030, 32032, 32760, 34320, 36036, 36960, 40040, 41184, 43680, 45045, 48048, 51480, 55440, 60060, 65520, 68640, 72072, 80080, 90090, 96096, 102960, 110880, 120120, 131040, 144144, 160160, 180180, 205920, 240240, 288288, 360360, 480480, 720720, 1441440.

2162160: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 33, 35, 36, 39, 40, 42, 44, 45, 48, 52, 54, 55, 56, 60, 63, 65, 66, 70, 72, 77, 78, 80, 84, 88, 90, 91, 99, 104, 105, 108, 110, 112, 117, 120, 126, 130, 132, 135, 140, 143, 144, 154, 156, 165, 168, 176, 180, 182, 189, 195, 198, 208, 210, 216, 220, 231, 234, 240, 252, 260, 264, 270, 273, 280, 286, 297, 308, 312, 315, 330, 336, 351, 360, 364, 378, 385, 390, 396, 420, 429, 432, 440, 455, 462, 468, 495, 504, 520, 528, 540, 546, 560, 572, 585, 594, 616, 624, 630, 660, 693, 702, 715, 720, 728, 756, 770, 780, 792, 819, 840, 858, 880, 910, 924, 936, 945, 990, 1001, 1008, 1040, 1080, 1092, 1144, 1155, 1170, 1188, 1232, 1260, 1287, 1320, 1365, 1386, 1404, 1430, 1456, 1485, 1512, 1540, 1560, 1584, 1638, 1680, 1716, 1755, 1820, 1848, 1872, 1890, 1980, 2002, 2079, 2145, 2160, 2184, 2288, 2310, 2340, 2376, 2457, 2520, 2574, 2640, 2730, 2772, 2808, 2860, 2970, 3003, 3024, 3080, 3120, 3276, 3432, 3465, 3510, 3640, 3696, 3780, 3861, 3960, 4004, 4095, 4158, 4290, 4368, 4620, 4680, 4752, 4914, 5005, 5040, 5148, 5460, 5544, 5616, 5720, 5940, 6006, 6160, 6435, 6552, 6864, 6930, 7020, 7280, 7560, 7722, 7920, 8008, 8190, 8316, 8580, 9009, 9240, 9360, 9828, 10010, 10296, 10395, 10920, 11088, 11440, 11880, 12012, 12285, 12870, 13104, 13860, 14040, 15015, 15120, 15444, 16016, 16380, 16632, 17160, 18018, 18480, 19305, 19656, 20020, 20592, 20790, 21840, 23760, 24024, 24570, 25740, 27027, 27720, 28080, 30030, 30888, 32760, 33264, 34320, 36036, 38610, 39312, 40040, 41580, 45045, 48048, 49140, 51480, 54054, 55440, 60060, 61776, 65520, 72072, 77220, 80080, 83160, 90090, 98280, 102960, 108108, 120120, 135135, 144144, 154440, 166320, 180180, 196560, 216216, 240240, 270270, 308880, 360360, 432432, 540540, 720720, 1081080, 2162160.

2882880: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 26, 28, 30, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 52, 55, 56, 60, 63, 64, 65, 66, 70, 72, 77, 78, 80, 84, 88, 90, 91, 96, 99, 104, 105, 110, 112, 117, 120, 126, 130, 132, 140, 143, 144, 154, 156, 160, 165, 168, 176, 180, 182, 192, 195, 198, 208, 210, 220, 224, 231, 234, 240, 252, 260, 264, 273, 280, 286, 288, 308, 312, 315, 320, 330, 336, 352, 360, 364, 385, 390, 396, 416, 420, 429, 440, 448, 455, 462, 468, 480, 495, 504, 520, 528, 546, 560, 572, 576, 585, 616, 624, 630, 660, 672, 693, 704, 715, 720, 728, 770, 780, 792, 819, 832, 840, 858, 880, 910, 924, 936, 960, 990, 1001, 1008, 1040, 1056, 1092, 1120, 1144, 1155, 1170, 1232, 1248, 1260, 1287, 1320, 1344, 1365, 1386, 1430, 1440, 1456, 1540, 1560, 1584, 1638, 1680, 1716, 1760, 1820, 1848, 1872, 1980, 2002, 2016, 2080, 2112, 2145, 2184, 2240, 2288, 2310, 2340, 2464, 2496, 2520, 2574, 2640, 2730, 2772, 2860, 2880, 2912, 3003, 3080, 3120, 3168, 3276, 3360, 3432, 3465, 3520, 3640, 3696, 3744, 3960, 4004, 4032, 4095, 4160, 4290, 4368, 4576, 4620, 4680, 4928, 5005, 5040, 5148, 5280, 5460, 5544, 5720, 5824, 6006, 6160, 6240, 6336, 6435, 6552, 6720, 6864, 6930, 7280, 7392, 7488, 7920, 8008, 8190, 8580, 8736, 9009, 9152, 9240, 9360, 10010, 10080, 10296, 10560, 10920, 11088, 11440, 12012, 12320, 12480, 12870, 13104, 13728, 13860, 14560, 14784, 15015, 15840, 16016, 16380, 17160, 17472, 18018, 18480, 18720, 20020, 20160, 20592, 21840, 22176, 22880, 24024, 24640, 25740, 26208, 27456, 27720, 29120, 30030, 31680, 32032, 32760, 34320, 36036, 36960, 37440, 40040, 41184, 43680, 44352, 45045, 45760, 48048, 51480, 52416, 55440, 60060, 64064, 65520, 68640, 72072, 73920, 80080, 82368, 87360, 90090, 96096, 102960, 110880, 120120, 131040, 137280, 144144, 160160, 180180, 192192, 205920, 221760, 240240, 262080, 288288, 320320, 360360, 411840, 480480, 576576, 720720, 960960, 1441440, 2882880.

3603600: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 28, 30, 33, 35, 36, 39, 40, 42, 44, 45, 48, 50, 52, 55, 56, 60, 63, 65, 66, 70, 72, 75, 77, 78, 80, 84, 88, 90, 91, 99, 100, 104, 105, 110, 112, 117, 120, 126, 130, 132, 140, 143, 144, 150, 154, 156, 165, 168, 175, 176, 180, 182, 195, 198, 200, 208, 210, 220, 225, 231, 234, 240, 252, 260, 264, 273, 275, 280, 286, 300, 308, 312, 315, 325, 330, 336, 350, 360, 364, 385, 390, 396, 400, 420, 429, 440, 450, 455, 462, 468, 495, 504, 520, 525, 528, 546, 550, 560, 572, 585, 600, 616, 624, 630, 650, 660, 693, 700, 715, 720, 728, 770, 780, 792, 819, 825, 840, 858, 880, 900, 910, 924, 936, 975, 990, 1001, 1008, 1040, 1050, 1092, 1100, 1144, 1155, 1170, 1200, 1232, 1260, 1287, 1300, 1320, 1365, 1386, 1400, 1430, 1456, 1540, 1560, 1575, 1584, 1638, 1650, 1680, 1716, 1800, 1820, 1848, 1872, 1925, 1950, 1980, 2002, 2100, 2145, 2184, 2200, 2275, 2288, 2310, 2340, 2475, 2520, 2574, 2600, 2640, 2730, 2772, 2800, 2860, 2925, 3003, 3080, 3120, 3150, 3276, 3300, 3432, 3465, 3575, 3600, 3640, 3696, 3850, 3900, 3960, 4004, 4095, 4200, 4290, 4368, 4400, 4550, 4620, 4680, 4950, 5005, 5040, 5148, 5200, 5460, 5544, 5720, 5775, 5850, 6006, 6160, 6300, 6435, 6552, 6600, 6825, 6864, 6930, 7150, 7280, 7700, 7800, 7920, 8008, 8190, 8400, 8580, 9009, 9100, 9240, 9360, 9900, 10010, 10296, 10725, 10920, 11088, 11440, 11550, 11700, 12012, 12600, 12870, 13104, 13200, 13650, 13860, 14300, 15015, 15400, 15600, 16016, 16380, 17160, 17325, 18018, 18200, 18480, 19800, 20020, 20475, 20592, 21450, 21840, 23100, 23400, 24024, 25025, 25200, 25740, 27300, 27720, 28600, 30030, 30800, 32175, 32760, 34320, 34650, 36036, 36400, 39600, 40040, 40950, 42900, 45045, 46200, 46800, 48048, 50050, 51480, 54600, 55440, 57200, 60060, 64350, 65520, 69300, 72072, 75075, 80080, 81900, 85800, 90090, 92400, 100100, 102960, 109200, 120120, 128700, 138600, 144144, 150150, 163800, 171600, 180180, 200200, 225225, 240240, 257400, 277200, 300300, 327600, 360360, 400400, 450450, 514800, 600600, 720720, 900900, 1201200, 1801800, 3603600.

4324320: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 52, 54, 55, 56, 60, 63, 65, 66, 70, 72, 77, 78, 80, 84, 88, 90, 91, 96, 99, 104, 105, 108, 110, 112, 117, 120, 126, 130, 132, 135, 140, 143, 144, 154, 156, 160, 165, 168, 176, 180, 182, 189, 195, 198, 208, 210, 216, 220, 224, 231, 234, 240, 252, 260, 264, 270, 273, 280, 286, 288, 297, 308, 312, 315, 330, 336, 351, 352, 360, 364, 378, 385, 390, 396, 416, 420, 429, 432, 440, 455, 462, 468, 480, 495, 504, 520, 528, 540, 546, 560, 572, 585, 594, 616, 624, 630, 660, 672, 693, 702, 715, 720, 728, 756, 770, 780, 792, 819, 840, 858, 864, 880, 910, 924, 936, 945, 990, 1001, 1008, 1040, 1056, 1080, 1092, 1120, 1144, 1155, 1170, 1188, 1232, 1248, 1260, 1287, 1320, 1365, 1386, 1404, 1430, 1440, 1456, 1485, 1512, 1540, 1560, 1584, 1638, 1680, 1716, 1755, 1760, 1820, 1848, 1872, 1890, 1980, 2002, 2016, 2079, 2080, 2145, 2160, 2184, 2288, 2310, 2340, 2376, 2457, 2464, 2520, 2574, 2640, 2730, 2772, 2808, 2860, 2912, 2970, 3003, 3024, 3080, 3120, 3168, 3276, 3360, 3432, 3465, 3510, 3640, 3696, 3744, 3780, 3861, 3960, 4004, 4095, 4158, 4290, 4320, 4368, 4576, 4620, 4680, 4752, 4914, 5005, 5040, 5148, 5280, 5460, 5544, 5616, 5720, 5940, 6006, 6048, 6160, 6240, 6435, 6552, 6864, 6930, 7020, 7280, 7392, 7560, 7722, 7920, 8008, 8190, 8316, 8580, 8736, 9009, 9240, 9360, 9504, 9828, 10010, 10080, 10296, 10395, 10920, 11088, 11232, 11440, 11880, 12012, 12285, 12320, 12870, 13104, 13728, 13860, 14040, 14560, 15015, 15120, 15444, 15840, 16016, 16380, 16632, 17160, 18018, 18480, 18720, 19305, 19656, 20020, 20592, 20790, 21840, 22176, 22880, 23760, 24024, 24570, 25740, 26208, 27027, 27720, 28080, 30030, 30240, 30888, 32032, 32760, 33264, 34320, 36036, 36960, 38610, 39312, 40040, 41184, 41580, 43680, 45045, 47520, 48048, 49140, 51480, 54054, 55440, 56160, 60060, 61776, 65520, 66528, 68640, 72072, 77220, 78624, 80080, 83160, 90090, 96096, 98280, 102960, 108108, 110880, 120120, 123552, 131040, 135135, 144144, 154440, 160160, 166320, 180180, 196560, 205920, 216216, 240240, 270270, 288288, 308880, 332640, 360360, 393120, 432432, 480480, 540540, 617760, 720720, 864864, 1081080, 1441440, 2162160, 4324320.

6486480: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 33, 35, 36, 39, 40, 42, 44, 45, 48, 52, 54, 55, 56, 60, 63, 65, 66, 70, 72, 77, 78, 80, 81, 84, 88, 90, 91, 99, 104, 105, 108, 110, 112, 117, 120, 126, 130, 132, 135, 140, 143, 144, 154, 156, 162, 165, 168, 176, 180, 182, 189, 195, 198, 208, 210, 216, 220, 231, 234, 240, 252, 260, 264, 270, 273, 280, 286, 297, 308, 312, 315, 324, 330, 336, 351, 360, 364, 378, 385, 390, 396, 405, 420, 429, 432, 440, 455, 462, 468, 495, 504, 520, 528, 540, 546, 560, 567, 572, 585, 594, 616, 624, 630, 648, 660, 693, 702, 715, 720, 728, 756, 770, 780, 792, 810, 819, 840, 858, 880, 891, 910, 924, 936, 945, 990, 1001, 1008, 1040, 1053, 1080, 1092, 1134, 1144, 1155, 1170, 1188, 1232, 1260, 1287, 1296, 1320, 1365, 1386, 1404, 1430, 1456, 1485, 1512, 1540, 1560, 1584, 1620, 1638, 1680, 1716, 1755, 1782, 1820, 1848, 1872, 1890, 1980, 2002, 2079, 2106, 2145, 2160, 2184, 2268, 2288, 2310, 2340, 2376, 2457, 2520, 2574, 2640, 2730, 2772, 2808, 2835, 2860, 2970, 3003, 3024, 3080, 3120, 3240, 3276, 3432, 3465, 3510, 3564, 3640, 3696, 3780, 3861, 3960, 4004, 4095, 4158, 4212, 4290, 4368, 4455, 4536, 4620, 4680, 4752, 4914, 5005, 5040, 5148, 5265, 5460, 5544, 5616, 5670, 5720, 5940, 6006, 6160, 6237, 6435, 6480, 6552, 6864, 6930, 7020, 7128, 7280, 7371, 7560, 7722, 7920, 8008, 8190, 8316, 8424, 8580, 8910, 9009, 9072, 9240, 9360, 9828, 10010, 10296, 10395, 10530, 10920, 11088, 11340, 11440, 11583, 11880, 12012, 12285, 12474, 12870, 13104, 13860, 14040, 14256, 14742, 15015, 15120, 15444, 16016, 16380, 16632, 16848, 17160, 17820, 18018, 18480, 19305, 19656, 20020, 20592, 20790, 21060, 21840, 22680, 23166, 23760, 24024, 24570, 24948, 25740, 27027, 27720, 28080, 29484, 30030, 30888, 31185, 32760, 33264, 34320, 35640, 36036, 36855, 38610, 39312, 40040, 41580, 42120, 45045, 45360, 46332, 48048, 49140, 49896, 51480, 54054, 55440, 57915, 58968, 60060, 61776, 62370, 65520, 71280, 72072, 73710, 77220, 80080, 81081, 83160, 84240, 90090, 92664, 98280, 99792, 102960, 108108, 115830, 117936, 120120, 124740, 135135, 144144, 147420, 154440, 162162, 166320, 180180, 185328, 196560, 216216, 231660, 240240, 249480, 270270, 294840, 308880, 324324, 360360, 405405, 432432, 463320, 498960, 540540, 589680, 648648, 720720, 810810, 926640, 1081080, 1297296, 1621620, 2162160, 3243240, 6486480.

7207200: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 28, 30, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 50, 52, 55, 56, 60, 63, 65, 66, 70, 72, 75, 77, 78, 80, 84, 88, 90, 91, 96, 99, 100, 104, 105, 110, 112, 117, 120, 126, 130, 132, 140, 143, 144, 150, 154, 156, 160, 165, 168, 175, 176, 180, 182, 195, 198, 200, 208, 210, 220, 224, 225, 231, 234, 240, 252, 260, 264, 273, 275, 280, 286, 288, 300, 308, 312, 315, 325, 330, 336, 350, 352, 360, 364, 385, 390, 396, 400, 416, 420, 429, 440, 450, 455, 462, 468, 480, 495, 504, 520, 525, 528, 546, 550, 560, 572, 585, 600, 616, 624, 630, 650, 660, 672, 693, 700, 715, 720, 728, 770, 780, 792, 800, 819, 825, 840, 858, 880, 900, 910, 924, 936, 975, 990, 1001, 1008, 1040, 1050, 1056, 1092, 1100, 1120, 1144, 1155, 1170, 1200, 1232, 1248, 1260, 1287, 1300, 1320, 1365, 1386, 1400, 1430, 1440, 1456, 1540, 1560, 1575, 1584, 1638, 1650, 1680, 1716, 1760, 1800, 1820, 1848, 1872, 1925, 1950, 1980, 2002, 2016, 2080, 2100, 2145, 2184, 2200, 2275, 2288, 2310, 2340, 2400, 2464, 2475, 2520, 2574, 2600, 2640, 2730, 2772, 2800, 2860, 2912, 2925, 3003, 3080, 3120, 3150, 3168, 3276, 3300, 3360, 3432, 3465, 3575, 3600, 3640, 3696, 3744, 3850, 3900, 3960, 4004, 4095, 4200, 4290, 4368, 4400, 4550, 4576, 4620, 4680, 4950, 5005, 5040, 5148, 5200, 5280, 5460, 5544, 5600, 5720, 5775, 5850, 6006, 6160, 6240, 6300, 6435, 6552, 6600, 6825, 6864, 6930, 7150, 7200, 7280, 7392, 7700, 7800, 7920, 8008, 8190, 8400, 8580, 8736, 8800, 9009, 9100, 9240, 9360, 9900, 10010, 10080, 10296, 10400, 10725, 10920, 11088, 11440, 11550, 11700, 12012, 12320, 12600, 12870, 13104, 13200, 13650, 13728, 13860, 14300, 14560, 15015, 15400, 15600, 15840, 16016, 16380, 16800, 17160, 17325, 18018, 18200, 18480, 18720, 19800, 20020, 20475, 20592, 21450, 21840, 22176, 22880, 23100, 23400, 24024, 25025, 25200, 25740, 26208, 26400, 27300, 27720, 28600, 30030, 30800, 31200, 32032, 32175, 32760, 34320, 34650, 36036, 36400, 36960, 39600, 40040, 40950, 41184, 42900, 43680, 45045, 46200, 46800, 48048, 50050, 50400, 51480, 54600, 55440, 57200, 60060, 61600, 64350, 65520, 68640, 69300, 72072, 72800, 75075, 79200, 80080, 81900, 85800, 90090, 92400, 93600, 96096, 100100, 102960, 109200, 110880, 114400, 120120, 128700, 131040, 138600, 144144, 150150, 160160, 163800, 171600, 180180, 184800, 200200, 205920, 218400, 225225, 240240, 257400, 277200, 288288, 300300, 327600, 343200, 360360, 400400, 450450, 480480, 514800, 554400, 600600, 655200, 720720, 800800, 900900, 1029600, 1201200, 1441440, 1801800, 2402400, 3603600, 7207200.

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32432400: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 33, 35, 36, 39, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 60, 63, 65, 66, 70, 72, 75, 77, 78, 80, 81, 84, 88, 90, 91, 99, 100, 104, 105, 108, 110, 112, 117, 120, 126, 130, 132, 135, 140, 143, 144, 150, 154, 156, 162, 165, 168, 175, 176, 180, 182, 189, 195, 198, 200, 208, 210, 216, 220, 225, 231, 234, 240, 252, 260, 264, 270, 273, 275, 280, 286, 297, 300, 308, 312, 315, 324, 325, 330, 336, 350, 351, 360, 364, 378, 385, 390, 396, 400, 405, 420, 429, 432, 440, 450, 455, 462, 468, 495, 504, 520, 525, 528, 540, 546, 550, 560, 567, 572, 585, 594, 600, 616, 624, 630, 648, 650, 660, 675, 693, 700, 702, 715, 720, 728, 756, 770, 780, 792, 810, 819, 825, 840, 858, 880, 891, 900, 910, 924, 936, 945, 975, 990, 1001, 1008, 1040, 1050, 1053, 1080, 1092, 1100, 1134, 1144, 1155, 1170, 1188, 1200, 1232, 1260, 1287, 1296, 1300, 1320, 1350, 1365, 1386, 1400, 1404, 1430, 1456, 1485, 1512, 1540, 1560, 1575, 1584, 1620, 1638, 1650, 1680, 1716, 1755, 1782, 1800, 1820, 1848, 1872, 1890, 1925, 1950, 1980, 2002, 2025, 2079, 2100, 2106, 2145, 2160, 2184, 2200, 2268, 2275, 2288, 2310, 2340, 2376, 2457, 2475, 2520, 2574, 2600, 2640, 2700, 2730, 2772, 2800, 2808, 2835, 2860, 2925, 2970, 3003, 3024, 3080, 3120, 3150, 3240, 3276, 3300, 3432, 3465, 3510, 3564, 3575, 3600, 3640, 3696, 3780, 3850, 3861, 3900, 3960, 4004, 4050, 4095, 4158, 4200, 4212, 4290, 4368, 4400, 4455, 4536, 4550, 4620, 4680, 4725, 4752, 4914, 4950, 5005, 5040, 5148, 5200, 5265, 5400, 5460, 5544, 5616, 5670, 5720, 5775, 5850, 5940, 6006, 6160, 6237, 6300, 6435, 6480, 6552, 6600, 6825, 6864, 6930, 7020, 7128, 7150, 7280, 7371, 7425, 7560, 7700, 7722, 7800, 7920, 8008, 8100, 8190, 8316, 8400, 8424, 8580, 8775, 8910, 9009, 9072, 9100, 9240, 9360, 9450, 9828, 9900, 10010, 10296, 10395, 10530, 10725, 10800, 10920, 11088, 11340, 11440, 11550, 11583, 11700, 11880, 12012, 12285, 12474, 12600, 12870, 13104, 13200, 13650, 13860, 14040, 14175, 14256, 14300, 14742, 14850, 15015, 15120, 15400, 15444, 15600, 16016, 16200, 16380, 16632, 16848, 17160, 17325, 17550, 17820, 18018, 18200, 18480, 18900, 19305, 19656, 19800, 20020, 20475, 20592, 20790, 21060, 21450, 21840, 22275, 22680, 23100, 23166, 23400, 23760, 24024, 24570, 24948, 25025, 25200, 25740, 26325, 27027, 27300, 27720, 28080, 28350, 28600, 29484, 29700, 30030, 30800, 30888, 31185, 32175, 32400, 32760, 33264, 34320, 34650, 35100, 35640, 36036, 36400, 36855, 37800, 38610, 39312, 39600, 40040, 40950, 41580, 42120, 42900, 44550, 45045, 45360, 46200, 46332, 46800, 48048, 49140, 49896, 50050, 51480, 51975, 52650, 54054, 54600, 55440, 56700, 57200, 57915, 58968, 59400, 60060, 61425, 61776, 62370, 64350, 65520, 69300, 70200, 71280, 72072, 73710, 75075, 75600, 77220, 80080, 81081, 81900, 83160, 84240, 85800, 89100, 90090, 92400, 92664, 96525, 98280, 99792, 100100, 102960, 103950, 105300, 108108, 109200, 113400, 115830, 117936, 118800, 120120, 122850, 124740, 128700, 135135, 138600, 140400, 144144, 147420, 150150, 154440, 155925, 162162, 163800, 166320, 171600, 178200, 180180, 184275, 185328, 193050, 196560, 200200, 207900, 210600, 216216, 225225, 226800, 231660, 240240, 245700, 249480, 257400, 270270, 277200, 289575, 294840, 300300, 308880, 311850, 324324, 327600, 356400, 360360, 368550, 386100, 400400, 405405, 415800, 421200, 432432, 450450, 463320, 491400, 498960, 514800, 540540, 579150, 589680, 600600, 623700, 648648, 675675, 720720, 737100, 772200, 810810, 831600, 900900, 926640, 982800, 1081080, 1158300, 1201200, 1247400, 1297296, 1351350, 1474200, 1544400, 1621620, 1801800, 2027025, 2162160, 2316600, 2494800, 2702700, 2948400, 3243240, 3603600, 4054050, 4633200, 5405400, 6486480, 8108100, 10810800, 16216200, 32432400.

36756720: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 27, 28, 30, 33, 34, 35, 36, 39, 40, 42, 44, 45, 48, 51, 52, 54, 55, 56, 60, 63, 65, 66, 68, 70, 72, 77, 78, 80, 84, 85, 88, 90, 91, 99, 102, 104, 105, 108, 110, 112, 117, 119, 120, 126, 130, 132, 135, 136, 140, 143, 144, 153, 154, 156, 165, 168, 170, 176, 180, 182, 187, 189, 195, 198, 204, 208, 210, 216, 220, 221, 231, 234, 238, 240, 252, 255, 260, 264, 270, 272, 273, 280, 286, 297, 306, 308, 312, 315, 330, 336, 340, 351, 357, 360, 364, 374, 378, 385, 390, 396, 408, 420, 429, 432, 440, 442, 455, 459, 462, 468, 476, 495, 504, 510, 520, 528, 540, 546, 560, 561, 572, 585, 594, 595, 612, 616, 624, 630, 660, 663, 680, 693, 702, 714, 715, 720, 728, 748, 756, 765, 770, 780, 792, 816, 819, 840, 858, 880, 884, 910, 918, 924, 935, 936, 945, 952, 990, 1001, 1008, 1020, 1040, 1071, 1080, 1092, 1105, 1122, 1144, 1155, 1170, 1188, 1190, 1224, 1232, 1260, 1287, 1309, 1320, 1326, 1360, 1365, 1386, 1404, 1428, 1430, 1456, 1485, 1496, 1512, 1530, 1540, 1547, 1560, 1584, 1638, 1680, 1683, 1716, 1755, 1768, 1785, 1820, 1836, 1848, 1870, 1872, 1890, 1904, 1980, 1989, 2002, 2040, 2079, 2142, 2145, 2160, 2184, 2210, 2244, 2288, 2295, 2310, 2340, 2376, 2380, 2431, 2448, 2457, 2520, 2574, 2618, 2640, 2652, 2730, 2772, 2805, 2808, 2856, 2860, 2970, 2992, 3003, 3024, 3060, 3080, 3094, 3120, 3213, 3276, 3315, 3366, 3432, 3465, 3510, 3536, 3570, 3640, 3672, 3696, 3740, 3780, 3861, 3927, 3960, 3978, 4004, 4080, 4095, 4158, 4284, 4290, 4368, 4420, 4488, 4590, 4620, 4641, 4680, 4752, 4760, 4862, 4914, 5005, 5040, 5049, 5148, 5236, 5304, 5355, 5460, 5544, 5610, 5616, 5712, 5720, 5940, 5967, 6006, 6120, 6160, 6188, 6426, 6435, 6545, 6552, 6630, 6732, 6864, 6930, 7020, 7140, 7280, 7293, 7344, 7480, 7560, 7722, 7735, 7854, 7920, 7956, 8008, 8190, 8316, 8415, 8568, 8580, 8840, 8976, 9009, 9180, 9240, 9282, 9360, 9520, 9724, 9828, 9945, 10010, 10098, 10296, 10395, 10472, 10608, 10710, 10920, 11088, 11220, 11440, 11781, 11880, 11934, 12012, 12155, 12240, 12285, 12376, 12852, 12870, 13090, 13104, 13260, 13464, 13860, 13923, 14040, 14280, 14586, 14960, 15015, 15120, 15444, 15470, 15708, 15912, 16016, 16065, 16380, 16632, 16830, 17017, 17136, 17160, 17680, 18018, 18360, 18480, 18564, 19305, 19448, 19635, 19656, 19890, 20020, 20196, 20592, 20790, 20944, 21420, 21840, 21879, 22440, 23205, 23562, 23760, 23868, 24024, 24310, 24570, 24752, 25245, 25704, 25740, 26180, 26520, 26928, 27027, 27720, 27846, 28080, 28560, 29172, 29835, 30030, 30888, 30940, 31416, 31824, 32130, 32760, 33264, 33660, 34034, 34320, 35343, 36036, 36465, 36720, 37128, 38610, 38896, 39270, 39312, 39780, 40040, 40392, 41580, 41769, 42840, 43758, 44880, 45045, 46410, 47124, 47736, 48048, 48620, 49140, 50490, 51051, 51408, 51480, 52360, 53040, 54054, 55440, 55692, 58344, 58905, 59670, 60060, 61776, 61880, 62832, 64260, 65520, 65637, 67320, 68068, 69615, 70686, 72072, 72930, 74256, 77220, 78540, 79560, 80080, 80784, 83160, 83538, 85085, 85680, 87516, 90090, 92820, 94248, 95472, 97240, 98280, 100980, 102102, 102960, 104720, 108108, 109395, 111384, 116688, 117810, 119340, 120120, 123760, 128520, 131274, 134640, 135135, 136136, 139230, 141372, 144144, 145860, 153153, 154440, 157080, 159120, 166320, 167076, 170170, 175032, 176715, 180180, 185640, 188496, 194480, 196560, 201960, 204204, 208845, 216216, 218790, 222768, 235620, 238680, 240240, 255255, 257040, 262548, 270270, 272272, 278460, 282744, 291720, 306306, 308880, 314160, 328185, 334152, 340340, 350064, 353430, 360360, 371280, 403920, 408408, 417690, 432432, 437580, 459459, 471240, 477360, 510510, 525096, 540540, 556920, 565488, 583440, 612612, 656370, 668304, 680680, 706860, 720720, 765765, 816816, 835380, 875160, 918918, 942480, 1021020, 1050192, 1081080, 1113840, 1225224, 1312740, 1361360, 1413720, 1531530, 1670760, 1750320, 1837836, 2042040, 2162160, 2297295, 2450448, 2625480, 2827440, 3063060, 3341520, 3675672, 4084080, 4594590, 5250960, 6126120, 7351344, 9189180, 12252240, 18378360, 36756720.

43243200: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 60, 63, 64, 65, 66, 70, 72, 75, 77, 78, 80, 84, 88, 90, 91, 96, 99, 100, 104, 105, 108, 110, 112, 117, 120, 126, 130, 132, 135, 140, 143, 144, 150, 154, 156, 160, 165, 168, 175, 176, 180, 182, 189, 192, 195, 198, 200, 208, 210, 216, 220, 224, 225, 231, 234, 240, 252, 260, 264, 270, 273, 275, 280, 286, 288, 297, 300, 308, 312, 315, 320, 325, 330, 336, 350, 351, 352, 360, 364, 378, 385, 390, 396, 400, 416, 420, 429, 432, 440, 448, 450, 455, 462, 468, 480, 495, 504, 520, 525, 528, 540, 546, 550, 560, 572, 576, 585, 594, 600, 616, 624, 630, 650, 660, 672, 675, 693, 700, 702, 704, 715, 720, 728, 756, 770, 780, 792, 800, 819, 825, 832, 840, 858, 864, 880, 900, 910, 924, 936, 945, 960, 975, 990, 1001, 1008, 1040, 1050, 1056, 1080, 1092, 1100, 1120, 1144, 1155, 1170, 1188, 1200, 1232, 1248, 1260, 1287, 1300, 1320, 1344, 1350, 1365, 1386, 1400, 1404, 1430, 1440, 1456, 1485, 1512, 1540, 1560, 1575, 1584, 1600, 1638, 1650, 1680, 1716, 1728, 1755, 1760, 1800, 1820, 1848, 1872, 1890, 1925, 1950, 1980, 2002, 2016, 2079, 2080, 2100, 2112, 2145, 2160, 2184, 2200, 2240, 2275, 2288, 2310, 2340, 2376, 2400, 2457, 2464, 2475, 2496, 2520, 2574, 2600, 2640, 2700, 2730, 2772, 2800, 2808, 2860, 2880, 2912, 2925, 2970, 3003, 3024, 3080, 3120, 3150, 3168, 3276, 3300, 3360, 3432, 3465, 3510, 3520, 3575, 3600, 3640, 3696, 3744, 3780, 3850, 3861, 3900, 3960, 4004, 4032, 4095, 4158, 4160, 4200, 4290, 4320, 4368, 4400, 4550, 4576, 4620, 4680, 4725, 4752, 4800, 4914, 4928, 4950, 5005, 5040, 5148, 5200, 5280, 5400, 5460, 5544, 5600, 5616, 5720, 5775, 5824, 5850, 5940, 6006, 6048, 6160, 6240, 6300, 6336, 6435, 6552, 6600, 6720, 6825, 6864, 6930, 7020, 7150, 7200, 7280, 7392, 7425, 7488, 7560, 7700, 7722, 7800, 7920, 8008, 8190, 8316, 8400, 8580, 8640, 8736, 8775, 8800, 9009, 9100, 9152, 9240, 9360, 9450, 9504, 9828, 9900, 10010, 10080, 10296, 10395, 10400, 10560, 10725, 10800, 10920, 11088, 11200, 11232, 11440, 11550, 11700, 11880, 12012, 12096, 12285, 12320, 12480, 12600, 12870, 13104, 13200, 13650, 13728, 13860, 14040, 14300, 14400, 14560, 14784, 14850, 15015, 15120, 15400, 15444, 15600, 15840, 16016, 16380, 16632, 16800, 17160, 17325, 17472, 17550, 17600, 18018, 18200, 18480, 18720, 18900, 19008, 19305, 19656, 19800, 20020, 20160, 20475, 20592, 20790, 20800, 21450, 21600, 21840, 22176, 22464, 22880, 23100, 23400, 23760, 24024, 24570, 24640, 25025, 25200, 25740, 26208, 26400, 27027, 27300, 27456, 27720, 28080, 28600, 29120, 29700, 30030, 30240, 30800, 30888, 31200, 31680, 32032, 32175, 32760, 33264, 33600, 34320, 34650, 35100, 36036, 36400, 36960, 37440, 37800, 38610, 39312, 39600, 40040, 40950, 41184, 41580, 42900, 43200, 43680, 44352, 45045, 45760, 46200, 46800, 47520, 48048, 49140, 50050, 50400, 51480, 51975, 52416, 52800, 54054, 54600, 55440, 56160, 57200, 59400, 60060, 60480, 61425, 61600, 61776, 62400, 64064, 64350, 65520, 66528, 68640, 69300, 70200, 72072, 72800, 73920, 75075, 75600, 77220, 78624, 79200, 80080, 81900, 82368, 83160, 85800, 87360, 90090, 92400, 93600, 95040, 96096, 96525, 98280, 100100, 100800, 102960, 103950, 108108, 109200, 110880, 112320, 114400, 118800, 120120, 122850, 123200, 123552, 128700, 131040, 133056, 135135, 137280, 138600, 140400, 144144, 145600, 150150, 151200, 154440, 157248, 158400, 160160, 163800, 166320, 171600, 180180, 184800, 187200, 192192, 193050, 196560, 200200, 205920, 207900, 216216, 218400, 221760, 225225, 228800, 237600, 240240, 245700, 247104, 257400, 262080, 270270, 277200, 280800, 288288, 300300, 302400, 308880, 320320, 327600, 332640, 343200, 360360, 369600, 386100, 393120, 400400, 411840, 415800, 432432, 436800, 450450, 475200, 480480, 491400, 514800, 540540, 554400, 561600, 576576, 600600, 617760, 655200, 665280, 675675, 686400, 720720, 772200, 786240, 800800, 831600, 864864, 900900, 960960, 982800, 1029600, 1081080, 1108800, 1201200, 1235520, 1310400, 1351350, 1441440, 1544400, 1601600, 1663200, 1729728, 1801800, 1965600, 2059200, 2162160, 2402400, 2702700, 2882880, 3088800, 3326400, 3603600, 3931200, 4324320, 4804800, 5405400, 6177600, 7207200, 8648640, 10810800, 14414400, 21621600, 43243200.

etc.

 

Prime Numbers Background:

This research is based off a posting I made a year ago about prime numbers and prime factorizations. At that same time, I was also exploring highly composite numbers.

Prime vs. Composite Numbers:

One way to think about prime vs. composite numbers is the following:

  • prime numbers = the atoms of numbers
  • composite numbers = the molecules of numbers

 

It Is All About Perspective:

To put this in perspective, most numbers have very few factors. Check out the factors of numbers from 18,446,744,073,709,551,615 (the largest 64-bit integer; 2^64-1) and downward:

18446744073709551615: 3, 5, 17, 257, 641, 65537, 6700417.
18446744073709551614: 2, 7, 7, 73, 127, 337, 92737, 649657.
18446744073709551613: 13, 3889, 364870227143809.
18446744073709551612: 2, 2, 3, 715827883, 2147483647.
18446744073709551611: 11, 59, 98818999, 287630261.
18446744073709551610: 2, 5, 23, 53301701, 1504703107.
18446744073709551609: 3, 3, 818923289, 2502845209.
18446744073709551608: 2, 2, 2, 2305843009213693951.
18446744073709551607: 7, 9241, 464773, 613566757.
18446744073709551606: 2, 3, 71, 42013, 1030686124187.
18446744073709551605: 5, 2551, 1446236305269271.
18446744073709551604: 2, 2, 37, 9902437, 12586817029.
18446744073709551603: 3, 139, 2306123, 19182323033.
18446744073709551602: 2, 157, 1973, 29775769179641.
18446744073709551601: 53, 348051774975651917.
18446744073709551600: 2, 2, 2, 2, 3, 3, 5, 5, 7, 11, 13, 31, 41, 61, 151, 331, 1321.
18446744073709551599: 19, 67, 14490765179661863.
18446744073709551598: 2, 17, 17, 17, 2927, 641387128649.
18446744073709551597: 3, 6148914691236517199.
18446744073709551596: 2, 2, 34421, 133978850655919.
...

See the next 5,000+ numbers in this sequence:

 

Performance / Application Versions:


Highly Composite Numbers v0.3

  • v0.1: November 14th, 2017:
    • A quick C# application and about an hour's worth of runtime shows the first results, up to 554,400.
  • v0.2: December 7th, 2018:
    • I improved my program significantly with a fiew optimizations, and could reproduce the previous hours' work in 15 seconds.
    • The main optimzation was to only search up until the square root of the number being factored, and pull out both factors. Such as, factoring 100 into 2 also gives you 50, and you only have to go as far as 10 -- the square root of 100. Note that 10 also gives you another 10, as 10 * 10 = 100, which you must ignore to avoid duplicate factors.
  • v0.3: December 8th, 2018:
    • "Graphical" and HTML output improvements, to help me blog the results.

 

Multi-Threading Optimzations?

Looking forward, I wish to get into multi-threading, it is something I have not coded for a long time. Not even our games were multi-threaded (to maintian simplicity), although they were highly optimized. The last time I coded multi-threaded algorigthms -- from memory -- was eScape. And that was in C++, helped by Jason Doucette.

 

Videos:


5060 and other Anti-Prime Nubmers - Numberphile

 

More:

Read about highly composite numbers:

See our other prototype work.

 

That is all.

 

 

About the Author: I am Matthew Doucette of Xona Games, an award-winning indie game studio that I founded with my twin brother. We make intensified arcade-style retro games. Our business, our games, our technology, and we as competitive gamers have won prestigious awards and received worldwide press. Our business has won $180,000 in contests. Our games have ranked from #1 in Canada to #1 in Japan, have become #1 best sellers in multiple countries, have won game contests, and have held 3 of the top 5 rated spots in Japan of all Xbox LIVE indie games. Our game engines have been awarded for technical excellence. And we, the developers, have placed #1 in competitive gaming competitions -- relating to the games we make. Read about our story, our awards, our games, and view our blog.



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