In the Vicinity of Cubic Numbers

In searching for properties of the number associated with my diurnal age, 26975, I noticed that the difference between this number and the nearest cubic number is 25 and that 25 divides 27000 (the nearest cubic number) to give 1080. I got to thinking about how many numbers, in the range up to 40,000, have this property.

Well, it turns out that 711 numbers do. A list of them is included at the end of this post. These numbers are not evenly distributed. They tend to cluster around cubic numbers with a large number of divisors and are sparsest around cubic numbers that are the cubes of primes. Figure 1 shows a list of the numbers from 1 to 35, together with their cubes and the numbers of both their divisors.

Figure 1: permalink

The graph of the 711 numbers is interesting, displaying a sinuous pattern. See Figure 2.

Figure 2: permalink

If we zoom in, things become clearer. Let's consider the cubic number 27000 that is the cube of 30. The former has 8 divisors and the latter 64. We'll look at the range from 26000 to 28000. Figure 3 shows the plot of the 94 numbers that qualify.

Figure 3: permalink

Notice how the graph is the same shape as that of \(y=x^3\). Next look at the cubic number 29791 that is the cube of 31. The former has four divisors (1, 31, 31 x 31 and 31 x 31 x 31) while the latter has two (1 and 31). The plot is shown in Figure 4 for the range from 28791 to 30791.

Figure 4: permalink

There are only six numbers and these are 28830, 29760, 29790, 29792, 29822, 30752. The numbers immediately preceding and succeeding the cubic number will always qualify since the difference is 1. Thus 26970 and 26972 are represented. Numbers with a difference of 31 will qualify and thus 26760 and 29822 are represented. Numbers with a difference of 31 x 31 will qualify and thus 28830 and 30752 are represented.

An interesting fact is that is that the difference not only divides the cubic number, it also always divides the number itself. For example, in the case 26975, the difference of 25 between 26975 and 27000 also divides 26975. To see why this is so, let's consider a cubic number \(c^3\) and a number \(n<c^3\) such that:
$$ 

\frac{c^3}{c^3-n} = k \text{ with integer }k>0\\

kc^3-kn =c^3\\

kn=kc^3-c^3\\

kn=c^3(k-1)\\

n=\dfrac{c^3}{k} (k-1) \\

n= \text{ difference } \times (k-1)
$$Hence \(n\) is always divisible by the difference. Take \(n=26975\) as an example. $$\frac{27000}{25}=1080\\26975=25 \times (1080-1)=25 \times 1079$$A similar proof can be concocted for the case where \(n>c^3\). Here is the list of the 711 numbers (permalink).

2, 4, 6, 7, 9, 10, 12, 18, 24, 26, 28, 30, 36, 48, 56, 60, 62, 63, 65, 66, 68, 72, 80, 100, 120, 124, 126, 130, 150, 180, 189, 192, 198, 204, 207, 208, 210, 212, 213, 214, 215, 217, 218, 219, 220, 222, 224, 225, 228, 234, 240, 243, 252, 270, 294, 336, 342, 344, 350, 392, 448, 480, 496, 504, 508, 510, 511, 513, 514, 516, 520, 528, 544, 576, 648, 702, 720, 726, 728, 730, 732, 738, 756, 810, 875, 900, 950, 960, 975, 980, 990, 992, 995, 996, 998, 999, 1001, 1002, 1004, 1005, 1008, 1010, 1020, 1025, 1040, 1050, 1100, 1125, 1210, 1320, 1330, 1332, 1342, 1452, 1536, 1584, 1620, 1632, 1656, 1664, 1674, 1680, 1692, 1696, 1701, 1704, 1710, 1712, 1716, 1719, 1720, 1722, 1724, 1725, 1726, 1727, 1729, 1730, 1731, 1732, 1734, 1736, 1737, 1740, 1744, 1746, 1752, 1755, 1760, 1764, 1776, 1782, 1792, 1800, 1824, 1836, 1872, 1920, 1944, 2028, 2184, 2196, 2198, 2210, 2366, 2548, 2646, 2688, 2695, 2716, 2730, 2736, 2737, 2740, 2742, 2743, 2745, 2746, 2748, 2751, 2752, 2758, 2772, 2793, 2800, 2842, 2940, 3150, 3240, 3250, 3300, 3330, 3348, 3350, 3360, 3366, 3370, 3372, 3374, 3376, 3378, 3380, 3384, 3390, 3400, 3402, 3420, 3450, 3500, 3510, 3600, 3840, 3968, 4032, 4064, 4080, 4088, 4092, 4094, 4095, 4097, 4098, 4100, 4104, 4112, 4128, 4160, 4224, 4352, 4624, 4896, 4912, 4914, 4930, 5202, 5508, 5589, 5616, 5670, 5724, 5751, 5760, 5778, 5796, 5805, 5808, 5814, 5820, 5823, 5824, 5826, 5828, 5829, 5830, 5831, 5833, 5834, 5835, 5836, 5838, 5840, 5841, 5844, 5850, 5856, 5859, 5868, 5886, 5904, 5913, 5940, 5994, 6048, 6075, 6156, 6318, 6498, 6840, 6858, 6860, 6878, 7220, 7500, 7600, 7680, 7750, 7800, 7840, 7875, 7900, 7920, 7936, 7950, 7960, 7968, 7975, 7980, 7984, 7990, 7992, 7995, 7996, 7998, 7999, 8001, 8002, 8004, 8005, 8008, 8010, 8016, 8020, 8025, 8032, 8040, 8050, 8064, 8080, 8100, 8125, 8160, 8200, 8250, 8320, 8400, 8500, 8820, 8918, 9072, 9114, 9198, 9212, 9234, 9240, 9252, 9254, 9258, 9260, 9262, 9264, 9268, 9270, 9282, 9288, 9310, 9324, 9408, 9450, 9604, 9702, 10164, 10406, 10527, 10560, 10604, 10626, 10637, 10640, 10644, 10646, 10647, 10649, 10650, 10652, 10656, 10659, 10670, 10692, 10736, 10769, 10890, 11132, 11638, 12144, 12166, 12168, 12190, 12696, 13056, 13248, 13312, 13392, 13440, 13536, 13568, 13608, 13632, 13680, 13696, 13716, 13728, 13752, 13760, 13770, 13776, 13788, 13792, 13797, 13800, 13806, 13808, 13812, 13815, 13816, 13818, 13820, 13821, 13822, 13823, 13825, 13826, 13827, 13828, 13830, 13832, 13833, 13836, 13840, 13842, 13848, 13851, 13856, 13860, 13872, 13878, 13888, 13896, 13920, 13932, 13952, 13968, 14016, 14040, 14080, 14112, 14208, 14256, 14336, 14400, 14592, 14688, 15000, 15500, 15600, 15620, 15624, 15626, 15630, 15650, 15750, 16250, 16900, 17238, 17407, 17472, 17524, 17550, 17563, 17568, 17572, 17574, 17575, 17577, 17578, 17580, 17584, 17589, 17602, 17628, 17680, 17745, 17914, 18252, 18954, 19440, 19602, 19656, 19674, 19680, 19682, 19684, 19686, 19692, 19710, 19764, 19926, 20412, 21168, 21266, 21504, 21560, 21609, 21728, 21756, 21840, 21854, 21888, 21896, 21903, 21920, 21924, 21936, 21938, 21944, 21945, 21948, 21950, 21951, 21953, 21954, 21956, 21959, 21960, 21966, 21968, 21980, 21984, 22001, 22008, 22016, 22050, 22064, 22148, 22176, 22295, 22344, 22400, 22638, 22736, 23548, 24360, 24388, 24390, 24418, 25230, 25875, 25920, 26000, 26100, 26250, 26325, 26400, 26460, 26500, 26550, 26625, 26640, 26700, 26730, 26750, 26775, 26784, 26800, 26820, 26850, 26865, 26875, 26880, 26892, 26900, 26910, 26925, 26928, 26940, 26946, 26950, 26955, 26960, 26964, 26970, 26973, 26975, 26976, 26980, 26982, 26985, 26988, 26990, 26991, 26992, 26994, 26995, 26996, 26997, 26998, 26999, 27001, 27002, 27003, 27004, 27005, 27006, 27008, 27009, 27010, 27012, 27015, 27018, 27020, 27024, 27025, 27027, 27030, 27036, 27040, 27045, 27050, 27054, 27060, 27072, 27075, 27090, 27100, 27108, 27120, 27125, 27135, 27150, 27180, 27200, 27216, 27225, 27250, 27270, 27300, 27360, 27375, 27450, 27500, 27540, 27600, 27675, 27750, 27900, 28000, 28080, 28125, 28350, 28830, 29760, 29790, 29792, 29822, 30752, 31744, 32256, 32512, 32640, 32704, 32736, 32752, 32760, 32764, 32766, 32767, 32769, 32770, 32772, 32776, 32784, 32800, 32832, 32896, 33024, 33280, 33792, 34606, 34848, 35574, 35640, 35816, 35838, 35904, 35910, 35926, 35928, 35934, 35936, 35938, 35940, 35946, 35948, 35964, 35970, 36036, 36058, 36234, 36300, 37026, 37268, 38148, 38726, 39015, 39168, 39236, 39270, 39287, 39296, 39300, 39302, 39303, 39305, 39306, 39308, 39312, 39321, 39338, 39372, 39440, 39593, 39882