12
$\begingroup$

There's a certain property that's shared between (as far as I know) infinite positive integers including 84, 96 and 108. Below are the first thousand numbers with this property; I added that many in case anyone would like to use a computer to find different properties they have in common.

84, 96, 108, 132, 150, 156, 198, 200, 204, 216, 
220, 228, 234, 260, 264, 272, 276, 300, 304, 306, 
312, 330, 340, 342, 348, 368, 372, 380, 390, 392, 
396, 408, 414, 420, 432, 440, 444, 450, 456, 460, 
464, 468, 476, 480, 490, 492, 496, 510, 516, 520, 
522, 528, 532, 540, 544, 552, 558, 564, 570, 580, 
588, 592, 600, 608, 612, 620, 624, 636, 644, 648, 
656, 660, 666, 672, 680, 684, 688, 690, 696, 708, 
714, 732, 735, 736, 738, 740, 744, 750, 752, 756, 
760, 765, 768, 774, 780, 792, 798, 804, 812, 816, 
820, 828, 846, 848, 852, 855, 860, 864, 868, 870, 
876, 880, 882, 888, 900, 912, 920, 924, 928, 930, 
936, 940, 944, 948, 952, 954, 966, 972, 976, 980, 
984, 990, 992, 996, 1000, 1012, 1020, 1026, 1032, 1035, 
1036, 1040, 1044, 1050, 1056, 1060, 1062, 1064, 1068, 1080, 
1092, 1098, 1104, 1110, 1116, 1125, 1128, 1140, 1148, 1160, 
1164, 1170, 1176, 1180, 1184, 1188, 1190, 1200, 1204, 1206, 
1212, 1215, 1218, 1220, 1224, 1230, 1236, 1240, 1242, 1248, 
1272, 1276, 1278, 1280, 1284, 1288, 1290, 1302, 1305, 1308, 
1312, 1314, 1316, 1320, 1330, 1332, 1340, 1350, 1356, 1360, 
1364, 1368, 1376, 1380, 1386, 1392, 1395, 1400, 1404, 1410, 
1416, 1420, 1422, 1428, 1449, 1450, 1452, 1460, 1464, 1470, 
1476, 1480, 1484, 1488, 1494, 1500, 1504, 1508, 1512, 1518, 
1520, 1524, 1530, 1540, 1548, 1550, 1554, 1560, 1566, 1572, 
1580, 1584, 1590, 1596, 1602, 1608, 1610, 1612, 1624, 1628, 
1632, 1638, 1640, 1644, 1650, 1652, 1656, 1660, 1665, 1668, 
1674, 1692, 1694, 1696, 1701, 1704, 1708, 1710, 1716, 1720, 
1722, 1736, 1740, 1746, 1750, 1752, 1760, 1764, 1770, 1776, 
1780, 1792, 1800, 1804, 1806, 1815, 1818, 1820, 1824, 1827, 
1830, 1836, 1840, 1845, 1848, 1850, 1854, 1860, 1872, 1876, 
1880, 1888, 1892, 1896, 1904, 1908, 1914, 1924, 1926, 1932, 
1935, 1936, 1940, 1944, 1952, 1953, 1960, 1962, 1968, 1974, 
1980, 1988, 1992, 1998, 2000, 2010, 2020, 2024, 2028, 2030, 
2034, 2040, 2044, 2046, 2050, 2052, 2058, 2060, 2064, 2068, 
2070, 2072, 2080, 2088, 2100, 2115, 2120, 2124, 2128, 2130, 
2132, 2136, 2140, 2142, 2144, 2150, 2160, 2170, 2175, 2178, 
2180, 2184, 2190, 2196, 2200, 2205, 2208, 2212, 2214, 2220, 
2226, 2232, 2236, 2244, 2250, 2256, 2260, 2262, 2272, 2280, 
2286, 2296, 2300, 2304, 2310, 2320, 2322, 2324, 2325, 2328, 
2331, 2332, 2336, 2340, 2350, 2352, 2358, 2360, 2366, 2368, 
2370, 2376, 2380, 2385, 2394, 2400, 2408, 2412, 2415, 2418, 
2420, 2424, 2430, 2436, 2440, 2442, 2444, 2448, 2460, 2466, 
2472, 2478, 2480, 2484, 2490, 2492, 2496, 2500, 2502, 2508, 
2516, 2528, 2530, 2535, 2538, 2540, 2541, 2544, 2550, 2552, 
2556, 2560, 2562, 2568, 2574, 2576, 2580, 2583, 2590, 2596, 
2600, 2604, 2610, 2616, 2620, 2624, 2625, 2628, 2632, 2640, 
2650, 2652, 2655, 2656, 2660, 2664, 2670, 2680, 2682, 2684, 
2700, 2704, 2706, 2709, 2712, 2716, 2718, 2720, 2728, 2730, 
2736, 2740, 2744, 2745, 2750, 2752, 2756, 2760, 2772, 2775, 
2780, 2784, 2788, 2790, 2808, 2814, 2820, 2826, 2828, 2832, 
2838, 2840, 2844, 2848, 2850, 2856, 2860, 2862, 2870, 2884, 
2886, 2898, 2900, 2904, 2910, 2916, 2920, 2924, 2928, 2934, 
2940, 2948, 2950, 2952, 2960, 2961, 2964, 2968, 2970, 2976, 
2980, 2982, 2988, 2992, 2996, 3000, 3006, 3008, 3010, 3015, 
3016, 3020, 3024, 3030, 3036, 3040, 3042, 3045, 3048, 3050, 
3052, 3060, 3066, 3068, 3072, 3075, 3078, 3080, 3087, 3090, 
3096, 3100, 3102, 3104, 3108, 3114, 3116, 3120, 3124, 3132, 
3140, 3144, 3150, 3160, 3164, 3168, 3172, 3180, 3186, 3190, 
3192, 3195, 3196, 3198, 3204, 3210, 3212, 3216, 3220, 3222, 
3224, 3225, 3232, 3240, 3248, 3255, 3256, 3258, 3260, 3264, 
3267, 3268, 3270, 3276, 3280, 3285, 3288, 3290, 3294, 3296, 
3300, 3304, 3312, 3318, 3320, 3330, 3336, 3339, 3340, 3344, 
3348, 3350, 3354, 3360, 3366, 3380, 3384, 3388, 3390, 3392, 
3402, 3408, 3410, 3416, 3420, 3424, 3430, 3432, 3438, 3440, 
3444, 3450, 3460, 3468, 3472, 3474, 3476, 3480, 3484, 3486, 
3488, 3492, 3498, 3500, 3504, 3510, 3525, 3528, 3536, 3540, 
3542, 3546, 3549, 3550, 3552, 3555, 3556, 3560, 3570, 3572, 
3576, 3580, 3582, 3584, 3588, 3600, 3604, 3608, 3612, 3616, 
3618, 3620, 3624, 3630, 3636, 3640, 3648, 3650, 3652, 3654, 
3660, 3663, 3666, 3668, 3672, 3680, 3690, 3692, 3696, 3700, 
3708, 3710, 3717, 3718, 3720, 3726, 3735, 3738, 3740, 3744, 
3750, 3752, 3760, 3762, 3768, 3770, 3774, 3776, 3780, 3784, 
3792, 3796, 3798, 3808, 3810, 3816, 3820, 3828, 3834, 3836, 
3840, 3843, 3848, 3852, 3860, 3864, 3870, 3872, 3876, 3880, 
3885, 3888, 3892, 3894, 3900, 3904, 3906, 3912, 3915, 3916, 
3920, 3924, 3930, 3936, 3940, 3942, 3948, 3950, 3952, 3960, 
3975, 3976, 3978, 3980, 3984, 3990, 3996, 4000, 4005, 4008, 
4012, 4014, 4020, 4026, 4028, 4030, 4040, 4048, 4056, 4059, 
4060, 4064, 4068, 4070, 4074, 4080, 4086, 4088, 4092, 4100, 
4104, 4108, 4110, 4116, 4120, 4122, 4128, 4130, 4134, 4136, 
4140, 4144, 4148, 4150, 4152, 4170, 4172, 4176, 4180, 4182, 
4185, 4192, 4194, 4200, 4212, 4220, 4221, 4228, 4230, 4235, 
4240, 4242, 4248, 4256, 4257, 4260, 4264, 4266, 4268, 4270, 
4272, 4280, 4284, 4288, 4290, 4296, 4300, 4302, 4305, 4312, 
4316, 4320, 4324, 4326, 4332, 4338, 4340, 4344, 4350, 4356, 
4360, 4365, 4368, 4374, 4380, 4384, 4386, 4392, 4396, 4410, 
4416, 4420, 4422, 4424, 4425, 4428, 4440, 4444, 4446, 4448, 
4450, 4452, 4460, 4464, 4466, 4470, 4472, 4473, 4482, 4484, 
4488, 4494, 4500, 4510, 4512, 4515, 4518, 4520, 4524, 4530, 
4532, 4536, 4540, 4544, 4545, 4554, 4556, 4560, 4563, 4564, 
4572, 4575, 4578, 4580, 4584, 4590, 4592, 4599, 4600, 4602, 
4608, 4620, 4624, 4626, 4628, 4632, 4635, 4636, 4640, 4644, 
4648, 4650, 4653, 4656, 4660, 4662, 4664, 4672, 4674, 4676, 
4680, 4686, 4690, 4692, 4698, 4700, 4704, 4708, 4710, 4716, 
4720, 4728, 4730, 4732, 4734, 4736, 4740, 4746, 4752, 4758, 
4760, 4768, 4770, 4774, 4776, 4780, 4784, 4788, 4794, 4796, 
4797, 4800, 4806, 4810, 4815, 4816, 4818, 4820, 4824, 4828, 
4830, 4832, 4836, 4840, 4842, 4844, 4848, 4850, 4860, 4872, 
4876, 4878, 4880, 4884, 4888, 4890, 4896, 4902, 4905, 4920, 
4932, 4935, 4940, 4944, 4950, 4956, 4960, 4964, 4968, 4970

The base the numbers are represented in does not matter, so there's no need to analyze the individual digits, nor to convert the numbers to another base. The property the numbers have in common is also purely mathematical, so don't worry about seven-segment displays, dates, letters or anything like that.

Hint:

The sequence is somewhat related to [A072510].(https://oeis.org/A072510)

Hint 2:

Multiply the first divisors.

$\endgroup$
9
  • $\begingroup$ Does "as far as I know" mean that you merely suspect (but do not know for certain) that there are infinitely many integers with the property? $\endgroup$
    – msh210
    Aug 21, 2023 at 18:45
  • $\begingroup$ I practically know, but I don't know how to prove it. $\endgroup$
    – Peter
    Aug 21, 2023 at 19:13
  • $\begingroup$ just to be sure: 84, 96 and 108 are the first(and smallest) numbers in the sequence, right? $\endgroup$
    – Novarg
    Aug 22, 2023 at 8:15
  • 1
    $\begingroup$ @Novarg Yes, that's correct. $\endgroup$
    – Peter
    Aug 22, 2023 at 10:37
  • 2
    $\begingroup$ @DmitryKamenetsky Sure! I might have made this puzzle a little too difficult - not because it's clever but because it's not good; it sort of requires an intuitive leap, which should be avoided. But I didn't want to delete it either, since at least a couple of people have worked very well to come closer to a solution. $\endgroup$
    – Peter
    Aug 25, 2023 at 13:53

2 Answers 2

4
$\begingroup$

Partial answer

Update: With the hint about "first k divisors", I found out the first 9 numbers satisfy being the product of 6th and 7th (smallest) divisors, however other numbers outside the list also have that property. Numbers with even number of divisors that are the product of their middle two divisors are the non-squares (pair up d and n/d).

Select[Range[1000], 
 DivisorSigma[0, #] >= 7 && Times @@ Divisors[#][[6 ;; 7]] == # &]
60, 72, 84, 90, 96, 108, 126, 132, 140, 150, 156, 160, 198, 200, 204, \
220, 224, 228, 234, 260, 276, 294, 306, 308, 315, 340, 342, 348, 350, \
352, 364, 372, 380, 392, 414, 416, 444, 460, 476, 486, 490, 492, 495, \
500, 516, 522, 525, 532, 544, 550, 558, 564, 572, 580, 585, 608, 620, \
636, 644, 650, 666, 675, 693, 708, 726, 732, 735, 736, 738, 740, 748, \
765, 774, 804, 812, 819, 820, 825, 836, 846, 850, 852, 855, 860, 868, \
876, 884, 928, 940, 948, 950, 954, 968, 975, 988, 992, 996}
Divisors[#][[6]]*Divisors[#][[7]] & /@ l
{84, 96, 108, 132, 150, 156, 198, 200, 204, 72, 220, 228, 234, 260, \
88, 578, 276, 60, 722, 306, 96, 110, 340, 342, 348, 1058, 372, 380, \
130, 392, 99, 96, 414, 42, 72, 110, 444, 90, 96, 460, 1682, 108, 476, \
48, 490, 492, 1922, 150, 516, 130, 522, 88, 532, 54, 544, 96, 558, \
564, 150, 580, 84, 2738, 48, 608, 108, 620, 96, 636, 644, 72, 3362, \
60, 666, 56, 170, 108, 3698, 150, 96, 708, 238, 732, 735, 736, 738, \
740, 96, 150, 4418, 63, 190, 765, 96, 774, 60, 72, 266, 804, 812, 96, \
820, 108, 846, 5618, 852, 855, 860, 72, 868, 150, 876, 110, 126, 96, \
54, 96, 200, 77, 928, 150, 72, 940, 6962, 948, 238, 954, 294, 108, \
7442, 140, 96, 90, 992, 996, 200, 1012, 60, 342, 96, 1035, 1036, 130, \
108, 70, 88, 1060, 1062, 266, 1068, 48, 84, 1098, 96, 150, 108, 1125, \
96, 60, 1148, 200, 1164, 90, 56, 1180, 1184, 99, 238, 48, 1204, 1206, \
1212, 1215, 294, 1220, 72, 150, 1236, 200, 414, 96, 96, 1276, 1278, \
160, 1284, 322, 150, 294, 1305, 1308, 1312, 1314, 1316, 48, 266, 108, \
1340, 90, 1356, 160, 1364, 72, 1376, 60, 99, 96, 1395, 80, 108, 150, \
96, 1420, 1422, 84, 1449, 1450, 132, 1460, 96, 70, 108, 200, 1484, \
96, 1494, 60, 1504, 1508, 56, 506, 160, 1524, 90, 110, 108, 1550, \
294, 48, 486, 1572, 1580, 72, 150, 84, 1602, 96, 322, 1612, 392, \
1628, 96, 117, 200, 1644, 110, 1652, 72, 1660, 1665, 1668, 486, 108, \
1694, 1696, 1701, 96, 1708, 90, 132, 200, 294, 392, 60, 1746, 350, \
96, 110, 63, 150, 96, 1780, 224, 48, 1804, 294, 1815, 1818, 130, 96, \
1827, 150, 108, 160, 1845, 56, 1850, 1854, 60, 72, 1876, 200, 1888, \
1892, 96, 224, 108, 638, 1924, 1926, 84, 1935, 352, 1940, 72, 1952, \
1953, 80, 1962, 96, 294, 54, 1988, 96, 486, 160, 150, 2020, 506, 156, \
406, 2034, 48, 2044, 682, 2050, 108, 294, 2060, 96, 2068, 90, 392, \
130, 72, 42, 2115, 200, 108, 224, 150, 2132, 96, 2140, 126, 2144, \
2150, 48, 434, 2175, 198, 2180, 56, 150, 108, 110, 315, 96, 2212, \
486, 60, 294, 72, 2236, 132, 90, 96, 2260, 754, 2272, 48, 2286, 392, \
460, 72, 70, 160, 486, 2324, 2325, 96, 2331, 2332, 2336, 54, 2350, \
56, 2358, 200, 2366, 1184, 150, 72, 140, 2385, 126, 48, 392, 108, \
483, 806, 220, 96, 90, 84, 200, 726, 2444, 72, 60, 2466, 96, 294, \
160, 108, 150, 2492, 96, 500, 2502, 132, 2516, 2528, 506, 2535, 486, \
2540, 2541, 96, 150, 638, 108, 160, 294, 96, 143, 224, 60, 2583, 490, \
2596, 130, 84, 90, 96, 2620, 1312, 525, 108, 392, 48, 2650, 156, \
2655, 2656, 140, 72, 150, 200, 2682, 2684, 54, 416, 726, 2709, 96, \
2716, 2718, 160, 682, 70, 72, 2740, 392, 2745, 550, 1376, 2756, 48, \
63, 2775, 2780, 96, 2788, 90, 72, 294, 60, 2826, 2828, 96, 726, 200, \
108, 2848, 150, 56, 143, 486, 490, 2884, 962, 126, 500, 88, 150, 108, \
200, 2924, 96, 2934, 42, 2948, 2950, 72, 160, 2961, 156, 392, 90, 96, \
2980, 294, 108, 272, 2996, 48, 3006, 1504, 490, 3015, 754, 3020, 56, \
150, 132, 160, 234, 609, 96, 3050, 3052, 54, 294, 3068, 96, 3075, \
342, 80, 3087, 150, 72, 500, 726, 3104, 84, 3114, 3116, 48, 3124, \
108, 3140, 96, 63, 200, 3164, 72, 3172, 60, 486, 638, 56, 3195, 3196, \
1014, 108, 150, 3212, 96, 140, 3222, 806, 3225, 3232, 48, 224, 651, \
814, 3258, 3260, 96, 3267, 3268, 150, 63, 160, 3285, 96, 490, 486, \
3296, 60, 392, 72, 294, 200, 90, 96, 3339, 3340, 304, 108, 3350, \
1014, 42, 187, 260, 72, 308, 150, 1696, 126, 96, 682, 392, 54, 3424, \
490, 88, 3438, 160, 84, 150, 3460, 204, 224, 3474, 3476, 48, 3484, \
294, 3488, 108, 726, 140, 96, 90, 3525, 56, 272, 60, 506, 3546, 3549, \
3550, 96, 3555, 3556, 200, 70, 3572, 96, 3580, 3582, 224, 156, 48, \
3604, 902, 84, 3616, 486, 3620, 96, 110, 108, 80, 96, 3650, 3652, \
126, 60, 3663, 1014, 3668, 72, 160, 90, 3692, 56, 500, 108, 490, \
3717, 3718, 48, 414, 3735, 294, 187, 72, 150, 392, 160, 198, 96, 754, \
1258, 1888, 42, 946, 96, 3796, 3798, 224, 150, 72, 3820, 132, 486, \
3836, 48, 3843, 962, 108, 3860, 56, 90, 352, 204, 200, 735, 72, 3892, \
726, 60, 1952, 126, 96, 783, 3916, 80, 108, 150, 96, 3940, 486, 84, \
3950, 304, 48, 3975, 392, 221, 3980, 96, 70, 108, 160, 4005, 96, \
4012, 4014, 60, 726, 4028, 806, 200, 352, 96, 4059, 140, 4064, 108, \
814, 294, 48, 4086, 392, 132, 500, 72, 4108, 150, 84, 200, 4122, 96, \
490, 1014, 968, 54, 224, 4148, 4150, 96, 150, 4172, 72, 209, 1394, \
837, 4192, 4194, 42, 108, 4220, 4221, 4228, 90, 4235, 160, 294, 72, \
224, 4257, 60, 1066, 486, 4268, 490, 96, 200, 63, 2048, 110, 96, 500, \
4302, 735, 154, 4316, 48, 4324, 294, 228, 4338, 140, 96, 150, 99, \
200, 4365, 56, 486, 60, 4384, 1462, 72, 4396, 63, 96, 221, 726, 392, \
4425, 108, 48, 4444, 234, 4448, 4450, 84, 4460, 72, 638, 150, 1118, \
4473, 486, 4484, 88, 294, 54, 902, 96, 735, 4518, 200, 156, 150, \
4532, 56, 4540, 2048, 4545, 198, 4556, 48, 4563, 4564, 108, 4575, \
294, 4580, 96, 90, 224, 4599, 200, 1014, 72, 42, 578, 4626, 4628, 96, \
4635, 4636, 160, 108, 392, 150, 4653, 96, 4660, 126, 968, 2048, 1558, \
4676, 48, 726, 490, 204, 486, 500, 56, 4708, 150, 108, 160, 96, 946, \
364, 4734, 1184, 60, 294, 72, 1014, 80, 4768, 90, 682, 96, 4780, 368, \
63, 1598, 4796, 4797, 48, 486, 962, 4815, 224, 726, 4820, 72, 4828, \
70, 4832, 156, 110, 4842, 4844, 96, 4850, 54, 56, 4876, 4878, 160, \
132, 1222, 150, 72, 1634, 4905, 48, 108, 735, 247, 96, 90, 84, 160, \
4964, 72, 490}

The number of divisors $\sigma_0(n)$ appears to be small, smooth numbers at least 10 with factors of 2 and 3, occasionally 5 and rarely 7. Yet some are excluded, like 60 which has $\sigma_0(60)=12$. I tried also looking at sum of divisors $\sigma_1(n)$ and sum of prime factors $\operatorname{sopfr}(n)$ (Plus @@ Times @@@ FactorInteger).

Here is the code to look for patterns where l is the numbers provided.

TableForm[Flatten[ConstantArray @@@ FactorInteger[#]] & /@ l]
2   2   3   7                           
2   2   2   2   2   3                   
2   2   3   3   3                       
2   2   3   11                          
2   3   5   5                           
2   2   3   13                          
2   3   3   11                          
2   2   2   5   5                       
2   2   3   17                          
2   2   2   3   3   3                   
2   2   5   11                          
2   2   3   19                          
2   3   3   13                          
2   2   5   13                          
2   2   2   3   11                      
2   2   2   2   17                      
2   2   3   23                          
2   2   3   5   5                       
2   2   2   2   19                      
2   3   3   17                          
2   2   2   3   13                      
2   3   5   11                          
2   2   5   17                          
2   3   3   19                          
2   2   3   29                          
2   2   2   2   23                      
2   2   3   31                          
2   2   5   19                          
2   3   5   13                          
2   2   2   7   7                       
2   2   3   3   11                      
2   2   2   3   17                      
2   3   3   23                          
2   2   3   5   7                       
2   2   2   2   3   3   3               
2   2   2   5   11                      
2   2   3   37                          
2   3   3   5   5                       
2   2   2   3   19                      
2   2   5   23                          
2   2   2   2   29                      
2   2   3   3   13                      
2   2   7   17                          
2   2   2   2   2   3   5               
2   5   7   7                           
2   2   3   41                          
2   2   2   2   31                      
2   3   5   17                          
2   2   3   43                          
2   2   2   5   13                      
2   3   3   29                          
2   2   2   2   3   11                  
2   2   7   19                          
2   2   3   3   3   5                   
2   2   2   2   2   17                  
2   2   2   3   23                      
2   3   3   31                          
2   2   3   47                          
2   3   5   19                          
2   2   5   29                          
2   2   3   7   7                       
2   2   2   2   37                      
2   2   2   3   5   5                   
2   2   2   2   2   19                  
2   2   3   3   17                      
2   2   5   31                          
2   2   2   2   3   13                  
2   2   3   53                          
2   2   7   23                          
2   2   2   3   3   3   3               
2   2   2   2   41                      
2   2   3   5   11                      
2   3   3   37                          
2   2   2   2   2   3   7               
2   2   2   5   17                      
2   2   3   3   19                      
2   2   2   2   43                      
2   3   5   23                          
2   2   2   3   29                      
2   2   3   59                          
2   3   7   17                          
2   2   3   61                          
3   5   7   7                           
2   2   2   2   2   23                  
2   3   3   41                          
2   2   5   37                          
2   2   2   3   31                      
2   3   5   5   5                       
2   2   2   2   47                      
2   2   3   3   3   7                   
2   2   2   5   19                      
3   3   5   17                          
2   2   2   2   2   2   2   2   3       
2   3   3   43                          
2   2   3   5   13                      
2   2   2   3   3   11                  
2   3   7   19                          
2   2   3   67                          
2   2   7   29                          
2   2   2   2   3   17                  
2   2   5   41                          
2   2   3   3   23                      
2   3   3   47                          
2   2   2   2   53                      
2   2   3   71                          
3   3   5   19                          
2   2   5   43                          
2   2   2   2   2   3   3   3           
2   2   7   31                          
2   3   5   29                          
2   2   3   73                          
2   2   2   2   5   11                  
2   3   3   7   7                       
2   2   2   3   37                      
2   2   3   3   5   5                   
2   2   2   2   3   19                  
2   2   2   5   23                      
2   2   3   7   11                      
2   2   2   2   2   29                  
2   3   5   31                          
2   2   2   3   3   13                  
2   2   5   47                          
2   2   2   2   59                      
2   2   3   79                          
2   2   2   7   17                      
2   3   3   53                          
2   3   7   23                          
2   2   3   3   3   3   3               
2   2   2   2   61                      
2   2   5   7   7                       
2   2   2   3   41                      
2   3   3   5   11                      
2   2   2   2   2   31                  
2   2   3   83                          
2   2   2   5   5   5                   
2   2   11  23                          
2   2   3   5   17                      
2   3   3   3   19                      
2   2   2   3   43                      
3   3   5   23                          
2   2   7   37                          
2   2   2   2   5   13                  
2   2   3   3   29                      
2   3   5   5   7                       
2   2   2   2   2   3   11              
2   2   5   53                          
2   3   3   59                          
2   2   2   7   19                      
2   2   3   89                          
2   2   2   3   3   3   5               
2   2   3   7   13                      
2   3   3   61                          
2   2   2   2   3   23                  
2   3   5   37                          
2   2   3   3   31                      
3   3   5   5   5                       
2   2   2   3   47                      
2   2   3   5   19                      
2   2   7   41                          
2   2   2   5   29                      
2   2   3   97                          
2   3   3   5   13                      
2   2   2   3   7   7                   
2   2   5   59                          
2   2   2   2   2   37                  
2   2   3   3   3   11                  
2   5   7   17                          
2   2   2   2   3   5   5               
2   2   7   43                          
2   3   3   67                          
2   2   3   101                         
3   3   3   3   3   5                   
2   3   7   29                          
2   2   5   61                          
2   2   2   3   3   17                  
2   3   5   41                          
2   2   3   103                         
2   2   2   5   31                      
2   3   3   3   23                      
2   2   2   2   2   3   13              
2   2   2   3   53                      
2   2   11  29                          
2   3   3   71                          
2   2   2   2   2   2   2   2   5       
2   2   3   107                         
2   2   2   7   23                      
2   3   5   43                          
2   3   7   31                          
3   3   5   29                          
2   2   3   109                         
2   2   2   2   2   41                  
2   3   3   73                          
2   2   7   47                          
2   2   2   3   5   11                  
2   5   7   19                          
2   2   3   3   37                      
2   2   5   67                          
2   3   3   3   5   5                   
2   2   3   113                         
2   2   2   2   5   17                  
2   2   11  31                          
2   2   2   3   3   19                  
2   2   2   2   2   43                  
2   2   3   5   23                      
2   3   3   7   11                      
2   2   2   2   3   29                  
3   3   5   31                          
2   2   2   5   5   7                   
2   2   3   3   3   13                  
2   3   5   47                          
2   2   2   3   59                      
2   2   5   71                          
2   3   3   79                          
2   2   3   7   17                      
3   3   7   23                          
2   5   5   29                          
2   2   3   11  11                      
2   2   5   73                          
2   2   2   3   61                      
2   3   5   7   7                       
2   2   3   3   41                      
2   2   2   5   37                      
2   2   7   53                          
2   2   2   2   3   31                  
2   3   3   83                          
2   2   3   5   5   5                   
2   2   2   2   2   47                  
2   2   13  29                          
2   2   2   3   3   3   7               
2   3   11  23                          
2   2   2   2   5   19                  
2   2   3   127                         
2   3   3   5   17                      
2   2   5   7   11                      
2   2   3   3   43                      
2   5   5   31                          
2   3   7   37                          
2   2   2   3   5   13                  
2   3   3   3   29                      
2   2   3   131                         
2   2   5   79                          
2   2   2   2   3   3   11              
2   3   5   53                          
2   2   3   7   19                      
2   3   3   89                          
2   2   2   3   67                      
2   5   7   23                          
2   2   13  31                          
2   2   2   7   29                      
2   2   11  37                          
2   2   2   2   2   3   17              
2   3   3   7   13                      
2   2   2   5   41                      
2   2   3   137                         
2   3   5   5   11                      
2   2   7   59                          
2   2   2   3   3   23                  
2   2   5   83                          
3   3   5   37                          
2   2   3   139                         
2   3   3   3   31                      
2   2   3   3   47                      
2   7   11  11                          
2   2   2   2   2   53                  
3   3   3   3   3   7                   
2   2   2   3   71                      
2   2   7   61                          
2   3   3   5   19                      
2   2   3   11  13                      
2   2   2   5   43                      
2   3   7   41                          
2   2   2   7   31                      
2   2   3   5   29                      
2   3   3   97                          
2   5   5   5   7                       
2   2   2   3   73                      
2   2   2   2   2   5   11              
2   2   3   3   7   7                   
2   3   5   59                          
2   2   2   2   3   37                  
2   2   5   89                          
2   2   2   2   2   2   2   2   7       
... 
TableForm[{#, DivisorSigma[0, #]} & /@ l]
84  12
96  12
108 12
132 12
150 12
156 12
198 12
200 12
204 12
216 16
220 12
228 12
234 12
260 12
264 16
272 10
276 12
300 18
304 10
306 12
312 16
330 16
340 12
342 12
348 12
368 10
372 12
380 12
390 16
392 12
396 18
408 16
414 12
420 24
432 20
440 16
444 12
450 18
456 16
460 12
464 10
468 18
476 12
480 24
490 12
492 12
496 10
510 16
516 12
520 16
522 12
528 20
532 12
540 24
544 12
552 16
558 12
564 12
570 16
580 12
588 18
592 10
600 24
608 12
612 18
620 12
624 20
636 12
644 12
648 20
656 10
660 24
666 12
672 24
680 16
684 18
688 10
690 16
696 16
708 12
714 16
732 12
735 12
736 12
738 12
740 12
744 16
750 16
752 10
756 24
760 16
765 12
768 18
774 12
780 24
792 24
798 16
804 12
812 12
816 20
820 12
828 18
846 12
848 10
852 12
855 12
860 12
864 24
868 12
870 16
876 12
880 20
882 18
888 16
900 27
912 20
920 16
924 24
928 12
930 16
936 24
940 12
944 10
948 12
952 16
954 12
966 16
972 18
976 10
980 18
984 16
990 24
992 12
996 12
1000    16
1012    12
1020    24
1026    16
1032    16
1035    12
1036    12
1040    20
1044    18
1050    24
1056    24
1060    12
1062    12
1064    16
1068    12
1080    32
1092    24
1098    12
1104    20
1110    16
1116    18
1125    12
1128    16
1140    24
1148    12
1160    16
1164    12
1170    24
1176    24
1180    12
1184    12
1188    24
1190    16
1200    30
1204    12
1206    12
1212    12
1215    12
1218    16
1220    12
1224    24
1230    16
1236    12
1240    16
1242    16
1248    24
1272    16
1276    12
1278    12
1280    18
1284    12
1288    16
1290    16
1302    16
1305    12
1308    12
1312    12
1314    12
1316    12
1320    32
1330    16
1332    18
1340    12
1350    24
1356    12
1360    20
1364    12
1368    24
1376    12
1380    24
1386    24
1392    20
1395    12
1400    24
1404    24
1410    16
1416    16
1420    12
1422    12
1428    24
1449    12
1450    12
1452    18
1460    12
1464    16
1470    24
1476    18
1480    16
1484    12
1488    20
1494    12
1500    24
1504    12
1508    12
1512    32
1518    16
1520    20
1524    12
1530    24
1540    24
1548    18
1550    12
1554    16
1560    32
1566    16
1572    12
1580    12
1584    30
1590    16
1596    24
1602    12
1608    16
1610    16
1612    12
1624    16
1628    12
1632    24
1638    24
1640    16
1644    12
1650    24
1652    12
1656    24
1660    12
1665    12
1668    12
1674    16
1692    18
1694    12
1696    12
1701    12
1704    16
1708    12
1710    24
1716    24
1720    16
1722    16
1736    16
1740    24
1746    12
1750    16
1752    16
1760    24
1764    27
1770    16
1776    20
1780    12
1792    18
1800    36
1804    12
1806    16
1815    12
1818    12
1820    24
1824    24
1827    12
1830    16
1836    24
1840    20
1845    12
1848    32
1850    12
1854    12
1860    24
1872    30
1876    12
1880    16
1888    12
1892    12
1896    16
1904    20
1908    18
1914    16
1924    12
1926    12
1932    24
1935    12
1936    15
1940    12
1944    24
1952    12
1953    12
1960    24
1962    12
1968    20
1974    16
1980    36
1988    12
1992    16
1998    16
2000    20
2010    16
2020    12
2024    16
2028    18
2030    16
2034    12
2040    32
2044    12
2046    16
2050    12
2052    24
2058    16
2060    12
2064    20
2068    12
2070    24
2072    16
2080    24
2088    24
2100    36
2115    12
2120    16
2124    18
2128    20
2130    16
2132    12
2136    16
2140    12
2142    24
2144    12
2150    12
2160    40
2170    16
2175    12
2178    18
2180    12
2184    32
2190    16
2196    18
2200    24
2205    18
2208    24
2212    12
2214    16
2220    24
2226    16
2232    24
2236    12
2244    24
2250    24
2256    20
2260    12
2262    16
2272    12
2280    32
2286    12
2296    16
2300    18
2304    27
2310    32
2320    20
2322    16
2324    12
2325    12
2328    16
2331    12
2332    12
2336    12
2340    36
2350    12
2352    30
2358    12
2360    16
2366    12
2368    14
2370    16
2376    32
2380    24
2385    12
2394    24
2400    36
2408    16
2412    18
2415    16
2418    16
2420    18
2424    16
2430    24
2436    24
2440    16
2442    16
2444    12
2448    30
2460    24
2466    12
2472    16
2478    16
2480    20
2484    24
2490    16
2492    12
2496    28
2500    15
2502    12
2508    24
2516    12
2528    12
2530    16
2535    12
2538    16
2540    12
2541    12
2544    20
2550    24
2552    16
2556    18
2560    20
2562    16
2568    16
2574    24
2576    20
2580    24
2583    12
2590    16
2596    12
2600    24
2604    24
2610    24
2616    16
2620    12
2624    14
2625    16
2628    18
2632    16
2640    40
2650    12
2652    24
2655    12
2656    12
2660    24
2664    24
2670    16
2680    16
2682    12
2684    12
2700    36
2704    15
2706    16
2709    12
2712    16
2716    12
2718    12
2720    24
2728    16
2730    32
2736    30
2740    12
2744    16
2745    12
2750    16
2752    14
2756    12
2760    32
2772    36
2775    12
2780    12
2784    24
2788    12
2790    24
2808    32
2814    16
2820    24
2826    12
2828    12
2832    20
2838    16
2840    16
2844    18
2848    12
2850    24
2856    32
2860    24
2862    16
2870    16
2884    12
2886    16
2898    24
2900    18
2904    24
2910    16
2916    21
2920    16
2924    12
2928    20
2934    12
2940    36
2948    12
2950    12
2952    24
2960    20
2961    12
2964    24
2968    16
2970    32
2976    24
2980    12
2982    16
2988    18
2992    20
2996    12
3000    32
3006    12
3008    14
3010    16
3015    12
3016    16
3020    12
3024    40
3030    16
3036    24
3040    24
3042    18
3045    16
3048    16
3050    12
3052    12
3060    36
3066    16
3068    12
3072    22
3075    12
3078    20
3080    32
3087    12
3090    16
3096    24
3100    18
3102    16
3104    12
3108    24
3114    12
3116    12
3120    40
3124    12
3132    24
3140    12
3144    16
3150    36
3160    16
3164    12
3168    36
3172    12
3180    24
3186    16
3190    16
3192    32
3195    12
3196    12
3198    16
3204    18
3210    16
3212    12
3216    20
3220    24
3222    12
3224    16
3225    12
3232    12
3240    40
3248    20
3255    16
3256    16
3258    12
3260    12
3264    28
3267    12
3268    12
3270    16
3276    36
3280    20
3285    12
3288    16
3290    16
3294    16
3296    12
3300    36
3304    16
3312    30
3318    16
3320    16
3330    24
3336    16
3339    12
3340    12
3344    20
3348    24
3350    12
3354    16
3360    48
3366    24
3380    18
3384    24
3388    18
3390    16
3392    14
3402    24
3408    20
3410    16
3416    16
3420    36
3424    12
3430    16
3432    32
3438    12
3440    20
3444    24
3450    24
3460    12
3468    18
3472    20
3474    12
3476    12
3480    32
3484    12
3486    16
3488    12
3492    18
3498    16
3500    24
3504    20
3510    32
3525    12
3528    36
3536    20
3540    24
3542    16
3546    12
3549    12
3550    12
3552    24
3555    12
3556    12
3560    16
3570    32
3572    12
3576    16
3580    12
3582    12
3584    20
3588    24
3600    45
3604    12
3608    16
3612    24
3616    12
3618    16
3620    12
3624    16
3630    24
3636    18
3640    32
3648    28
3650    12
3652    12
3654    24
3660    24
3663    12
3666    16
3668    12
3672    32
3680    24
3690    24
3692    12
3696    40
3700    18
3708    18
3710    16
3717    12
3718    12
3720    32
3726    20
3735    12
3738    16
3740    24
3744    36
3750    20
3752    16
3760    20
3762    24
3768    16
3770    16
3774    16
3776    14
3780    48
3784    16
3792    20
3796    12
3798    12
3808    24
3810    16
3816    24
3820    12
3828    24
3834    16
3836    12
3840    36
3843    12
3848    16
3852    18
3860    12
3864    32
3870    24
3872    18
3876    24
3880    16
3885    16
3888    30
3892    12
3894    16
3900    36
3904    14
3906    24
3912    16
3915    16
3916    12
3920    30
3924    18
3930    16
3936    24
3940    12
3942    16
3948    24
3950    12
3952    20
3960    48
3975    12
3976    16
3978    24
3980    12
3984    20
3990    32
3996    24
4000    24
4005    12
4008    16
4012    12
4014    12
4020    24
4026    16
4028    12
4030    16
4040    16
4048    20
4056    24
4059    12
4060    24
4064    12
4068    18
4070    16
4074    16
4080    40
4086    12
4088    16
4092    24
4100    18
4104    32
4108    12
4110    16
4116    24
4120    16
4122    12
4128    24
4130    16
4134    16
4136    16
4140    36
4144    20
4148    12
4150    12
4152    16
4170    16
4172    12
4176    30
4180    24
4182    16
4185    16
4192    12
4194    12
4200    48
4212    30
4220    12
4221    12
4228    12
4230    24
4235    12
4240    20
4242    16
4248    24
4256    24
4257    12
4260    24
4264    16
4266    16
4268    12
4270    16
4272    20
4280    16
4284    36
4288    14
4290    32
4296    16
4300    18
4302    12
4305    16
4312    24
4316    12
4320    48
4324    12
4326    16
4332    18
4338    12
4340    24
4344    16
4350    24
4356    27
4360    16
4365    12
4368    40
4374    16
4380    24
4384    12
4386    16
4392    24
4396    12
4410    36
4416    28
4420    24
4422    16
4424    16
4425    12
4428    24
4440    32
4444    12
4446    24
4448    12
4450    12
4452    24
4460    12
4464    30
4466    16
4470    16
4472    16
4473    12
4482    16
4484    12
4488    32
4494    16
4500    36
4510    16
4512    24
4515    16
4518    12
4520    16
4524    24
4530    16
4532    12
4536    40
4540    12
4544    14
4545    12
4554    24
4556    12
4560    40
4563    12
4564    12
4572    18
4575    12
4578    16
4580    12
4584    16
4590    32
4592    20
4599    12
4600    24
4602    16
4608    30
4620    48
4624    15
4626    12
4628    12
4632    16
4635    12
4636    12
4640    24
4644    24
4648    16
4650    24
4653    12
4656    20
4660    12
4662    24
4664    16
4672    14
4674    16
4676    12
4680    48
4686    16
4690    16
4692    24
4698    20
4700    18
4704    36
4708    12
4710    16
4716    18
4720    20
4728    16
4730    16
4732    18
4734    12
4736    16
4740    24
4746    16
4752    40
4758    16
4760    32
4768    12
4770    24
4774    16
4776    16
4780    12
4784    20
4788    36
4794    16
4796    12
4797    12
4800    42
4806    16
4810    16
4815    12
4816    20
4818    16
4820    12
4824    24
4828    12
4830    32
4832    12
4836    24
4840    24
4842    12
4844    12
4848    20
4850    12
4860    36
4872    32
4876    12
4878    12
4880    20
4884    24
4888    16
4890    16
4896    36
4902    16
4905    12
4920    32
4932    18
4935    16
4940    24
4944    20
4950    36
4956    24
4960    24
4964    12
4968    32
4970    16
```
$\endgroup$
4
  • 1
    $\begingroup$ That's a very important observation! I suspect this will be quite useful for later answers. $\endgroup$
    – Peter
    Aug 22, 2023 at 5:22
  • $\begingroup$ A possible observation: in certain sequences where (at least it appears) the factorisation p_0*p_1*p_3*p’ works for any prime p > some amount (for example, 2*2*5*p’ for p’ > 10), possibly excluding the case p’ = p_n. In those cases it appears the “some amount” is the product p_0p_n. For example, I would conjecture that other than 3*5*11*11, the next such number that fits this category in the sequence is 3*5*11*37. This would indicate that the rule may correlate its largest prime with the others (though this may just be a coincidence, as there are exceptions). $\endgroup$
    – yanjunk
    Aug 23, 2023 at 3:59
  • $\begingroup$ Note also that while the 4-2 and 5-2 sequences both start at p' = 17 (presumably after 2^4), the 6-2 and 7-2 sequences start at 37 (presumably after 2^5). $\endgroup$
    – yanjunk
    Aug 23, 2023 at 4:11
  • 1
    $\begingroup$ A rule very close to making sense is that one of the two middle factors (which multiply to the number) is a power of a prime. The rule might involve the two middle factors in some way. $\endgroup$
    – yanjunk
    Aug 23, 2023 at 4:44
1
$\begingroup$

Partial answer:

Numbers that don't divide the product of their first six factors, nor first half of factors (minus one) is pretty close. For reference, the sequence is:

84, 96, 108, 132, 150, 156, 198, 200, 204, 220, 228, 234, 260, 272, 276, 294, 304, 306, 312, 340, 342, 348, 368, 372, 380, 392, 408, 414, 444, 456, 460

Code:

from functools import reduce
import operator

def factors(n):    
    return sorted(reduce(list.__add__, 
                ([i, n//i] for i in range(1, int(n**0.5) + 1) if n % i == 0)))
def prod(a, i):
    return reduce(operator.mul, a[0:i], 1)

thing = [factors(i) for i in range(2, 5000)]
newset = set()
for a in thing:
    if prod(a, max(5, len(a) // 2 - 1)) % a[-1] != 0:
        newset.add(a[-1])

In particular, if you only consider the first 6 divisors, the set returned seems to be a superset of the list in the question. Of particular interest is the counterexample 294, which has 12 divisors and like 84 is divisible by the first 6 but not the first 5.

The code is the above but replacing the max(5, ...) with 5.

$\endgroup$
5
  • $\begingroup$ Note that twelve of the entries in the list have only 10 divisors. They of the form $2^4\cdot p$ where $p$ runs over the primes between $17$ and $61$ (one might speculate that they are the primes between $2^4$ and $2^6$). $\endgroup$ Aug 28, 2023 at 2:11
  • $\begingroup$ I don't underatand what you mean by "don't divide the product of their first half of factors (minus one)"; I haven't been able to recreate your results. $\endgroup$ Aug 28, 2023 at 2:13
  • $\begingroup$ Apologies. I flubbed in explaining it, and have edited the answer to include the code and fix the explanation. $\endgroup$
    – yanjunk
    Aug 28, 2023 at 2:57
  • $\begingroup$ @GregMartin I forgot to mention you in my reply. Hopefully this isn't considered rude (to make a comment solely to mention you). $\endgroup$
    – yanjunk
    Aug 28, 2023 at 3:17
  • $\begingroup$ Btw you can use sympy for simple number theory functions. $\endgroup$
    – qwr
    Aug 28, 2023 at 21:18

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