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Old 2021-10-03, 13:57   #1
alpertron
 
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Aug 2002
Buenos Aires, Argentina

22·5·71 Posts
Default Computing divisors from complete factorization

I would like to generate the divisors of a number from its complete factorization in powers of prime numbers.

It is very easy to generate all of them if I do not need to be sorted, but I would like to show them in ascending order.

Is there any way to do show the complete list of divisors without generating all of them and then sorting them?

The number may have millions of divisors (this is the case for primorials and factorials).

The idea is to show the list of divisors in batches of up to 1000 divisors. The user can continue asking for more divisors or cancel the generation of the list.

Last fiddled with by alpertron on 2021-10-03 at 14:12
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Old 2021-10-03, 16:05   #2
a1call
 
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"Rashid Naimi"
Oct 2015
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Pari-gp code:
Code:
fordiv(5!, X, print(X))

Code:
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120

Quote:
fordiv(n, X, seq)

Evaluates seq, where the formal variable X ranges through the divisors of n (see divisors, which is used as a subroutine). It is assumed that factor can handle n, without negative exponents. Instead of n, it is possible to input a factorization matrix, i.e. the output of factor(n).

This routine uses divisors as a subroutine, then loops over the divisors. In particular, if n is an integer, divisors are sorted by increasing size.

To avoid storing all divisors, possibly using a lot of memory, the following (slower) routine loops over the divisors using essentially constant space:

Code:
 FORDIV(N)=
  { my(F = factor(N), P = F[,1], E = F[,2]);
  
    forvec(v = vector(#E, i, [0,E[i]]), X = factorback(P, v));
  }
  ? for(i=1, 10^6, FORDIV(i))
  time = 11,180 ms.
  ? for(i=1, 10^6, fordiv(i, d, ))
  time = 2,667 ms.
https://pari.math.u-bordeaux.fr/dochtml/html-stable/

Last fiddled with by a1call on 2021-10-03 at 16:14
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Old 2021-10-04, 00:23   #3
alpertron
 
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Aug 2002
Buenos Aires, Argentina

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Thanks, but it does not help me.

As a test, I entered:

Code:
FORDIV(N)=
  { my(F = factor(N), P = F[,1], E = F[,2]);
  
    forvec(v = vector(#E, i, [0,E[i]]), print(factorback(P, v)));
  }
Then gp shows the list of divisors of 36 as:

Code:
(21:19) gp > print(FORDIV(36));
1
3
9
2
6
18
4
12
36
0
Apart from the extra zero, the divisors are not sorted in ascending order as I need.
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Old 2021-10-04, 00:47   #4
a1call
 
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"Rashid Naimi"
Oct 2015
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You can use fordiv() to get sorted results but you will be limited by amount of memory that can store the divisors. If you use the memory limiting function then you will have to sort the results yourself. If the divisors exceed the allocated memory, then you would need to use external files and things will be very slow with Pari.

So if you can fit all the divisors in the memory there are no issues:

Code:
fordiv(36, X, print(X))
Code:
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36
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Old 2021-10-04, 00:56   #5
alpertron
 
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From my first post, you can see that I cannot sort the divisors.

There may be millions of divisors, but I have to generate the smallest 1000 of them. After the user presses a button, the program has to show the next 1000 divisors and so on.

I cannot sort the divisors because there is no memory to hold the complete list.
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Old 2021-10-04, 01:05   #6
a1call
 
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"Rashid Naimi"
Oct 2015
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Knowing all the prime factors and their recurrence cannot give you sorted divisors without sorting or brute force for general/arbitrary cases.

Not sure what scenario would not have any memory at all. If you could store 1000 integers and if time is not a factor you could go through all the "millions" of divisors and only store 1st, then 2nd, ... smallest 1000 in the range you find, each time.

Last fiddled with by a1call on 2021-10-04 at 01:07
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Old 2021-10-04, 01:24   #7
alpertron
 
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Ok. Let's use an example: the number is 100! I know its complete factorization.

I want to show the first 1000 divisors, then the next 1000 divisors and so on.
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Old 2021-10-04, 03:03   #8
a1call
 
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I am a slow coder, but to get my point across here is a dirty, inefficient and incomplete code.

Pari-gp code:
Code:

\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
factorialFace=23
theregister=100
theThou = vector(theregister+1);

FORDIVSort(N,lowerRange)=
  { my(F = factor(N), P = F[,1], E = F[,2]);
    theCounter=0;
    forvec(v = vector(#E, i, [0,E[i]]), a=factorback(P, v);
    if(a>lowerRange,
        theCounter=theCounter+1;
        if(theCounter>theregister+1,theCounter=theregister+1;theThou=vecsort(theThou););
        theThou[theCounter]=a;
        
    );
    \\print(a);
    );
    print("***********");
    print(theThou);
    return(theThou[theregister]);
  }



theLow=1
while(theLow<2*factorialFace!,{
    for(i=1,#theThou ,theThou [i]=1);
    theLow=FORDIVSort(factorialFace!,theLow);
})

Code:
***********
[2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 60, 63, 64, 65, 66, 68, 69, 70, 72, 75, 76, 77, 78, 80, 81, 84, 85, 88, 90, 91, 92, 95, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 115, 117, 119, 120, 121, 125, 126, 128, 130, 132, 133, 135, 136, 138, 140, 143, 144, 147, 25852016738884976640000]
***********
[150, 152, 153, 154, 156, 160, 161, 162, 165, 168, 170, 171, 175, 176, 180, 182, 184, 187, 189, 190, 192, 195, 196, 198, 200, 204, 207, 208, 209, 210, 216, 220, 221, 224, 225, 228, 230, 231, 234, 238, 240, 242, 243, 245, 247, 250, 252, 253, 255, 256, 260, 264, 266, 270, 272, 273, 275, 276, 280, 285, 286, 288, 294, 297, 299, 300, 304, 306, 308, 312, 315, 320, 322, 323, 324, 325, 330, 336, 340, 342, 343, 345, 350, 351, 352, 357, 360, 363, 364, 368, 374, 375, 378, 380, 384, 385, 390, 391, 392, 396, 25852016738884976640000]
***********
[399, 400, 405, 408, 414, 416, 418, 420, 425, 429, 432, 437, 440, 441, 442, 448, 450, 455, 456, 459, 460, 462, 468, 475, 476, 480, 483, 484, 486, 490, 494, 495, 500, 504, 506, 510, 512, 513, 520, 525, 528, 532, 539, 540, 544, 546, 550, 552, 560, 561, 567, 570, 572, 575, 576, 585, 588, 594, 595, 598, 600, 605, 608, 612, 616, 621, 624, 625, 627, 630, 637, 640, 644, 646, 648, 650, 660, 663, 665, 672, 675, 680, 684, 686, 690, 693, 700, 702, 704, 714, 715, 720, 726, 728, 729, 735, 736, 741, 748, 750, 25852016738884976640000]
***********
[756, 759, 760, 765, 768, 770, 780, 782, 784, 792, 798, 800, 805, 810, 816, 819, 825, 828, 832, 833, 836, 840, 847, 850, 855, 858, 864, 874, 875, 880, 882, 884, 891, 896, 897, 900, 910, 912, 918, 920, 924, 931, 935, 936, 945, 950, 952, 960, 966, 968, 969, 972, 975, 980, 988, 990, 1000, 1001, 1008, 1012, 1020, 1024, 1026, 1029, 1035, 1040, 1045, 1050, 1053, 1056, 1064, 1071, 1078, 1080, 1088, 1089, 1092, 1100, 1104, 1105, 1120, 1122, 1125, 1127, 1134, 1140, 1144, 1150, 1152, 1155, 1170, 1173, 1176, 1188, 1190, 1196, 1197, 1200, 1210, 1215, 25852016738884976640000]
***********
[1216, 1224, 1225, 1232, 1235, 1242, 1248, 1250, 1254, 1260, 1265, 1274, 1275, 1280, 1287, 1288, 1292, 1296, 1300, 1309, 1311, 1320, 1323, 1326, 1330, 1344, 1350, 1360, 1365, 1368, 1372, 1375, 1377, 1380, 1386, 1400, 1404, 1408, 1425, 1428, 1430, 1440, 1449, 1452, 1456, 1458, 1463, 1470, 1472, 1482, 1485, 1495, 1496, 1500, 1512, 1518, 1520, 1530, 1536, 1539, 1540, 1547, 1560, 1564, 1568, 1573, 1575, 1584, 1596, 1600, 1610, 1615, 1617, 1620, 1625, 1632, 1638, 1650, 1656, 1664, 1666, 1672, 1680, 1683, 1694, 1700, 1701, 1710, 1715, 1716, 1725, 1728, 1729, 1748, 1750, 1755, 1760, 1764, 1768, 1771, 25852016738884976640000]
***********
[1782, 1785, 1792, 1794, 1800, 1815, 1820, 1824, 1836, 1840, 1848, 1862, 1863, 1870, 1872, 1875, 1881, 1890, 1900, 1904, 1911, 1920, 1925, 1932, 1936, 1938, 1944, 1950, 1955, 1960, 1976, 1980, 1989, 1995, 2000, 2002, 2016, 2024, 2025, 2040, 2048, 2052, 2057, 2058, 2070, 2079, 2080, 2090, 2093, 2100, 2106, 2112, 2125, 2128, 2142, 2145, 2156, 2160, 2176, 2178, 2184, 2185, 2187, 2200, 2205, 2208, 2210, 2223, 2240, 2244, 2250, 2254, 2261, 2268, 2275, 2277, 2280, 2288, 2295, 2299, 2300, 2304, 2310, 2340, 2346, 2352, 2375, 2376, 2380, 2392, 2394, 2400, 2415, 2420, 2430, 2431, 2432, 2448, 2450, 2457, 25852016738884976640000]
***********
[2464, 2470, 2475, 2484, 2496, 2499, 2500, 2508, 2520, 2530, 2541, 2548, 2550, 2560, 2565, 2574, 2576, 2584, 2592, 2600, 2618, 2622, 2625, 2640, 2646, 2652, 2660, 2673, 2688, 2691, 2695, 2700, 2717, 2720, 2730, 2736, 2737, 2744, 2750, 2754, 2760, 2772, 2783, 2793, 2800, 2805, 2808, 2816, 2835, 2850, 2856, 2860, 2875, 2880, 2898, 2904, 2907, 2912, 2916, 2925, 2926, 2940, 2944, 2964, 2970, 2975, 2990, 2992, 3000, 3003, 3024, 3025, 3036, 3040, 3059, 3060, 3072, 3078, 3080, 3087, 3094, 3105, 3120, 3128, 3135, 3136, 3146, 3150, 3159, 3168, 3185, 3192, 3200, 3213, 3220, 3230, 3234, 3240, 3250, 3264, 25852016738884976640000]
***********
[3267, 3276, 3289, 3300, 3312, 3315, 3325, 3328, 3332, 3344, 3360, 3366, 3375, 3381, 3388, 3400, 3402, 3420, 3430, 3432, 3450, 3456, 3458, 3465, 3496, 3500, 3510, 3519, 3520, 3528, 3536, 3542, 3553, 3564, 3570, 3575, 3584, 3588, 3591, 3600, 3630, 3640, 3645, 3648, 3672, 3675, 3680, 3696, 3705, 3724, 3726, 3740, 3744, 3750, 3762, 3773, 3780, 3795, 3800, 3808, 3822, 3825, 3840, 3850, 3861, 3864, 3872, 3876, 3888, 3900, 3910, 3920, 3927, 3933, 3952, 3960, 3969, 3978, 3990, 4000, 4004, 4025, 4032, 4048, 4050, 4080, 4095, 4096, 4104, 4114, 4116, 4125, 4131, 4140, 4158, 4160, 4165, 4180, 4186, 4199, 25852016738884976640000]
***********
[4200, 4212, 4224, 4235, 4250, 4256, 4275, 4284, 4290, 4301, 4312, 4320, 4347, 4352, 4356, 4368, 4370, 4374, 4375, 4389, 4400, 4410, 4416, 4420, 4446, 4455, 4459, 4480, 4485, 4488, 4500, 4508, 4522, 4536, 4550, 4554, 4560, 4576, 4590, 4598, 4600, 4608, 4617, 4620, 4641, 4655, 4675, 4680, 4692, 4704, 4719, 4725, 4750, 4752, 4760, 4784, 4788, 4800, 4807, 4830, 4840, 4845, 4851, 4860, 4862, 4864, 4875, 4896, 4900, 4914, 4928, 4940, 4950, 4968, 4992, 4998, 5000, 5005, 5016, 5040, 5049, 5060, 5082, 5083, 5096, 5100, 5103, 5120, 5130, 5145, 5148, 5152, 5168, 5175, 5184, 5187, 5200, 5225, 5236, 5244, 25852016738884976640000]
***********
[5250, 5265, 5280, 5292, 5304, 5313, 5320, 5346, 5355, 5376, 5382, 5390, 5400, 5434, 5440, 5445, 5460, 5472, 5474, 5488, 5500, 5508, 5520, 5525, 5544, 5566, 5586, 5589, 5600, 5610, 5616, 5625, 5632, 5635, 5643, 5670, 5681, 5700, 5712, 5720, 5733, 5750, 5760, 5775, 5796, 5808, 5814, 5824, 5831, 5832, 5850, 5852, 5865, 5880, 5888, 5928, 5929, 5940, 5950, 5967, 5980, 5984, 5985, 6000, 6006, 6048, 6050, 6072, 6075, 6080, 6118, 6120, 6125, 6144, 6156, 6160, 6171, 6174, 6175, 6188, 6210, 6237, 6240, 6256, 6270, 6272, 6279, 6292, 6300, 6318, 6325, 6336, 6370, 6375, 6384, 6400, 6426, 6435, 6440, 6460, 25852016738884976640000]
***********
[6468, 6480, 6500, 6517, 6528, 6534, 6545, 6552, 6555, 6561, 6578, 6600, 6615, 6624, 6630, 6650, 6656, 6664, 6669, 6688, 6720, 6732, 6750, 6762, 6776, 6783, 6800, 6804, 6825, 6831, 6840, 6860, 6864, 6875, 6885, 6897, 6900, 6912, 6916, 6930, 6992, 7000, 7007, 7020, 7038, 7040, 7056, 7072, 7084, 7106, 7125, 7128, 7140, 7150, 7168, 7176, 7182, 7200, 7245, 7260, 7280, 7290, 7293, 7296, 7315, 7344, 7350, 7360, 7371, 7392, 7410, 7425, 7429, 7448, 7452, 7475, 7480, 7488, 7497, 7500, 7524, 7546, 7560, 7590, 7600, 7616, 7623, 7644, 7650, 7680, 7695, 7700, 7722, 7728, 7735, 7744, 7752, 7776, 7800, 7820, 25852016738884976640000]
***********
[7840, 7854, 7865, 7866, 7875, 7889, 7904, 7920, 7938, 7956, 7980, 8000, 8008, 8019, 8050, 8064, 8073, 8075, 8085, 8096, 8100, 8125, 8151, 8160, 8190, 8192, 8208, 8211, 8228, 8232, 8250, 8262, 8280, 8316, 8320, 8330, 8349, 8360, 8372, 8379, 8398, 8400, 8415, 8424, 8448, 8470, 8500, 8505, 8512, 8550, 8568, 8575, 8580, 8602, 8624, 8625, 8640, 8645, 8694, 8704, 8712, 8721, 8736, 8740, 8748, 8750, 8775, 8778, 8800, 8820, 8832, 8840, 8855, 8892, 8910, 8918, 8925, 8960, 8970, 8976, 9000, 9009, 9016, 9044, 9072, 9075, 9100, 9108, 9120, 9152, 9163, 9177, 9180, 9196, 9200, 9216, 9234, 9240, 9261, 9282, 25852016738884976640000]
***********
[9310, 9315, 9350, 9360, 9384, 9405, 9408, 9438, 9450, 9477, 9500, 9504, 9520, 9555, 9568, 9576, 9600, 9614, 9625, 9639, 9660, 9680, 9690, 9702, 9720, 9724, 9728, 9750, 9775, 9792, 9800, 9801, 9828, 9856, 9867, 9880, 9900, 9936, 9945, 9975, 9984, 9996, 10000, 10010, 10032, 10080, 10098, 10120, 10125, 10143, 10164, 10166, 10192, 10200, 10206, 10240, 10241, 10260, 10285, 10290, 10296, 10304, 10336, 10350, 10368, 10374, 10395, 10400, 10450, 10465, 10472, 10488, 10500, 10530, 10557, 10560, 10584, 10608, 10625, 10626, 10640, 10659, 10692, 10710, 10725, 10752, 10764, 10773, 10780, 10800, 10829, 10868, 10880, 10890, 10920, 10925, 10935, 10944, 10948, 10976, 25852016738884976640000]
***********
[11000, 11011, 11016, 11025, 11040, 11050, 11088, 11115, 11132, 11172, 11178, 11200, 11220, 11232, 11250, 11264, 11270, 11286, 11305, 11319, 11340, 11362, 11375, 11385, 11400, 11424, 11440, 11466, 11475, 11495, 11500, 11520, 11550, 11583, 11592, 11616, 11628, 11648, 11662, 11664, 11700, 11704, 11730, 11760, 11776, 11781, 11799, 11856, 11858, 11875, 11880, 11900, 11907, 11934, 11960, 11968, 11970, 12000, 12012, 12075, 12096, 12100, 12103, 12144, 12150, 12155, 12160, 12236, 12240, 12250, 12285, 12288, 12312, 12320, 12342, 12348, 12350, 12375, 12376, 12393, 12397, 12420, 12474, 12480, 12495, 12512, 12540, 12544, 12558, 12584, 12597, 12600, 12636, 12650, 12672, 12705, 12740, 12750, 12768, 12800, 25852016738884976640000]
***********
[12825, 12852, 12870, 12880, 12903, 12920, 12936, 12960, 13000, 13034, 13041, 13056, 13068, 13090, 13104, 13110, 13122, 13125, 13156, 13167, 13200, 13230, 13248, 13260, 13300, 13312, 13328, 13338, 13365, 13376, 13377, 13440, 13455, 13464, 13475, 13500, 13524, 13552, 13566, 13585, 13600, 13608, 13650, 13662, 13680, 13685, 13720, 13728, 13750, 13770, 13794, 13800, 13824, 13832, 13851, 13860, 13915, 13923, 13965, 13984, 14000, 14014, 14025, 14040, 14076, 14080, 14112, 14144, 14157, 14168, 14175, 14212, 14250, 14256, 14280, 14300, 14336, 14352, 14364, 14375, 14399, 14400, 14421, 14490, 14520, 14535, 14553, 14560, 14580, 14586, 14592, 14625, 14630, 14651, 14688, 14700, 14720, 14742, 14784, 14820, 25852016738884976640000]
***********
[14850, 14858, 14875, 14896, 14904, 14950, 14960, 14976, 14994, 15000, 15015, 15048, 15092, 15120, 15125, 15147, 15180, 15200, 15232, 15246, 15249, 15288, 15295, 15300, 15309, 15360, 15390, 15400, 15435, 15444, 15456, 15470, 15488, 15504, 15525, 15552, 15561, 15600, 15640, 15675, 15680, 15708, 15730, 15732, 15750, 15778, 15795, 15808, 15827, 15840, 15876, 15912, 15925, 15939, 15960, 16000, 16016, 16038, 16065, 16093, 16100, 16128, 16146, 16150, 16170, 16192, 16200, 16250, 16302, 16320, 16335, 16380, 16384, 16416, 16422, 16445, 16456, 16464, 16500, 16524, 16560, 16575, 16625, 16632, 16640, 16660, 16698, 16720, 16744, 16758, 16767, 16796, 16800, 16830, 16848, 16875, 16896, 16905, 16929, 16940, 25852016738884976640000]
***********
[17000, 17010, 17017, 17024, 17043, 17100, 17136, 17150, 17160, 17199, 17204, 17248, 17250, 17280, 17290, 17325, 17388, 17408, 17424, 17442, 17472, 17480, 17493, 17496, 17500, 17550, 17556, 17595, 17600, 17640, 17664, 17680, 17710, 17765, 17784, 17787, 17820, 17836, 17850, 17875, 17901, 17920, 17940, 17952, 17955, 18000, 18018, 18032, 18088, 18144, 18150, 18200, 18216, 18225, 18240, 18304, 18326, 18354, 18360, 18375, 18392, 18400, 18432, 18468, 18480, 18513, 18522, 18525, 18564, 18620, 18630, 18700, 18711, 18720, 18768, 18810, 18816, 18837, 18865, 18876, 18900, 18954, 18975, 19000, 19008, 19019, 19040, 19110, 19125, 19136, 19152, 19159, 19200, 19228, 19250, 19278, 19305, 19320, 19360, 19380, 25852016738884976640000]
***********
[19404, 19440, 19448, 19456, 19481, 19500, 19550, 19551, 19584, 19600, 19602, 19635, 19656, 19665, 19683, 19712, 19734, 19760, 19800, 19845, 19872, 19890, 19950, 19968, 19992, 20000, 20007, 20020, 20064, 20125, 20160, 20196, 20240, 20250, 20286, 20328, 20332, 20349, 20384, 20400, 20412, 20475, 20480, 20482, 20493, 20520, 20570, 20580, 20592, 20608, 20625, 20655, 20672, 20691, 20700, 20736, 20748, 20790, 20800, 20825, 20900, 20930, 20944, 20976, 20995, 21000, 21021, 21060, 21114, 21120, 21168, 21175, 21216, 21250, 21252, 21280, 21318, 21375, 21384, 21413, 21420, 21450, 21504, 21505, 21528, 21546, 21560, 21600, 21658, 21735, 21736, 21760, 21780, 21840, 21850, 21870, 21879, 21888, 21896, 21945, 25852016738884976640000]
***********
[21952, 22000, 22022, 22032, 22050, 22080, 22100, 22113, 22176, 22230, 22264, 22275, 22287, 22295, 22344, 22356, 22400, 22425, 22440, 22464, 22491, 22500, 22528, 22540, 22572, 22610, 22638, 22680, 22724, 22750, 22770, 22800, 22848, 22869, 22880, 22932, 22950, 22990, 23000, 23023, 23040, 23085, 23100, 23166, 23184, 23205, 23232, 23256, 23275, 23296, 23324, 23328, 23375, 23400, 23408, 23460, 23520, 23552, 23562, 23595, 23598, 23625, 23667, 23712, 23716, 23750, 23760, 23800, 23814, 23868, 23920, 23936, 23940, 24000, 24024, 24035, 24057, 24150, 24192, 24200, 24206, 24219, 24225, 24255, 24288, 24300, 24310, 24320, 24375, 24453, 24472, 24480, 24500, 24570, 24576, 24624, 24633, 24640, 24684, 24696, 25852016738884976640000]
***********
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***********
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***********
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***********
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***********
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***********
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***********
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***********
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***********
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***********
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***********
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***********
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***********
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***********
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***********
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***********
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***********
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***********
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....

Last fiddled with by a1call on 2021-10-04 at 03:04
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Old 2021-10-04, 03:15   #9
alpertron
 
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All numbers between 1 and 100 are divisors of 100! but it appears that there are missing divisors in your list.

Last fiddled with by alpertron on 2021-10-04 at 03:15
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Old 2021-10-04, 03:18   #10
a1call
 
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"Rashid Naimi"
Oct 2015
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I chose 23! not 100!. It takes too long to go through all the divisors. You can try changing the factorialFace variable to 100 if you are willing to wait ? period of time.
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Old 2021-10-04, 03:24   #11
slandrum
 
Jan 2021
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Quote:
Originally Posted by alpertron View Post
All numbers between 1 and 100 are divisors of 100! but it appears that there are missing divisors in your list.
If you look at the code, it's not doing 100!, it's doing 23! - I assume it's just an example piece of code. However, it does leave out the factor 1.

edit: And by the time I looked at the code to see what it did and typed up a reply, it was already replied to.

Last fiddled with by slandrum on 2021-10-04 at 03:26
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