27872: Another Palindromic Number

For the last 800 days they've been coming around every 100 days and today marks yet another palindromic day as I turn 27872 days old. I've mentioned one of the properties of this number in my post titled Difference of Two Cubic Numbers. I noted that this palindrome is a difference of two cubes:$$27872=38^3-30^3$$Another property of this palindrome is that it is the sum of two prime palindromes in two different ways:$$ \begin{align} 27872 &= 11311 + 16561\\ &=12421 + 15451 \end{align}$$In the range up to 40000, there are only 25 palindromes with this property and these are the initial terms of OEIS A356854:

282, 484, 858, 888, 21912, 22722, 23832, 24642, 25752, 26662, 26762, 26862, 26962, 27672, 27772, 27872, 27972, 28482, 28782, 28882, 28982, 29692, 29792, 29892, 29992

27872 also has the property that it is the smallest palindrome with exactly seven prime factors, counted with multiplicity. This qualifies it for membership in OEIS A076886: smallest palindrome with exactly \(n\) prime factors (counted with multiplicity). See Figure 1 where the initial terms with their \(n\) values are listed.

Figure 1

In 100 days I'll be able to celebrate 27972 that's also a member of OEIS A356854. After that, it is only 31 days to the next palindromic day: 30003.

Difference of Two Cubic Numbers

I'm surprised that I've not dealt with this topic before but as far as I can tell I haven't. The topic in question is numbers that are the difference of two cubes, or more specifically the difference of two positive cubes. My diurnal age today is 27872, a palindrome, with the property that:$$27872 = 38^3-30^3$$It's easy enough to develop an algorithm to determine all such numbers in the range up to 40000 and the total is 825. However, if we consider only those numbers equal to or greater than 27872, then only 188 numbers qualify. They are (permalink):

27872, 27937, 28063, 28415, 28460, 28519, 28568, 28656, 28672, 28701, 28737, 28791, 28828, 28854, 29051, 29062, 29078, 29080, 29107, 29279, 29393, 29402, 29448, 29528, 29575, 29617, 29666, 29701, 29727, 29735, 29763, 29764, 29783, 29790, 30016, 30024, 30043, 30105, 30248, 30301, 30312, 30483, 30571, 30708, 30807, 30907, 30970, 31024, 31031, 31040, 31085, 31106, 31213, 31228, 31232, 31304, 31437, 31519, 31768, 31806, 31841, 31869, 31976, 32039, 32137, 32227, 32256, 32319, 32425, 32445, 32464, 32465, 32552, 32562, 32643, 32704, 32741, 32760, 32761, 32767, 32832, 32851, 32858, 32920, 32949, 32984, 33077, 33193, 33336, 33391, 33472, 33614, 33724, 33740, 33752, 33875, 34027, 34047, 34209, 34391, 34489, 34531, 34606, 34658, 34669, 34784, 34875, 34902, 34930, 34937, 35008, 35028, 35163, 35189, 35208, 35315, 35317, 35425, 35576, 35594, 35721, 35812, 35873, 35910, 35929, 35936, 35971, 36008, 36016, 36153, 36253, 36297, 36316, 36504, 36506, 36560, 36631, 36632, 36785, 36829, 37000, 37043, 37107, 37296, 37297, 37367, 37395, 37448, 37449, 37576, 37648, 37962, 37969, 37973, 38017, 38142, 38151, 38285, 38304, 38402, 38486, 38528, 38575, 38619, 38647, 38656, 38779, 38792, 38961, 39004, 39088, 39130, 39179, 39240, 39247, 39277, 39296, 39303, 39331, 39368, 39500, 39611, 39636, 39797, 39807, 39815, 39816, 39823 (see OEIS A181123)

Of these 188, there are five numbers that can be expressed as a difference of two cubes in more that one way. These are 27937, 28063, 34209, 35208 and 35929. The details are as follows:$$ \begin{align} 27937 &= 33^3- 20^3 \\ &=97^3- 96^3 \\ 28063 &= 31^3- 12^3 \\ &=40^3- 33^3 \\34209 &= 33^3- 12^3 \\ &=40^3- 31^3 \\ 35208 &= 33^3- 9^3 \\ &= 34^3- 16^3 \\ 35929 &= 33^3- 2^3 \\ &=34^3- 15^3 \end{align}$$Notice that 27937 is a difference of successive cubes but, because it is not prime, it cannot be a Cuban prime. The only Cuban prime in the range is 33391where:$$33391=106^3-105^3$$I've dealt with this category of primes in my blog post titled Cuban Primes way back in July of 2016.

Sums and Concatenations of Cubes and Squares

There's something very obvious about the number associated with my diurnal age today. The number is 27064 and the cubes (27 and 64) stand out clearly. In fact 27064 can be written as a sum of two cubes:$$ \begin{align} 27064 &=27000+64\\&=30^3+4^3 \end{align}$$Unfortunately, the number cannot be written as a concatenation of two cubes because the zero gets in the way. The problem is that 4 cubed has only two digits. However, the cubes of the numbers from 5 to 9 all have three digits and so the zero disappears. This allows us to write the following numbers as both sums and concatenations of two cubes. The symbol | indicates concatenation$$ \begin{align} 27125 =30^3+5^3 = 3^3|5^3\\27216 =30^3+6^3 = 3^3|6^3\\27343 = 30^3+7^3 = 3^3|7^3\\27512 = 30^3+ 8^3 = 3^3|8^3\\27729 = 30^3+9^3=3^3|9^3 \end{align} $$This series of numbers is the last that will occur in my lifetime because the next such sets of numbers will begin with 64125. However, if we were to consider sums of squares and concatenations of squares then I may see these come to pass. Consider the following sets of numbers, some of which occur more than once (permalink).$$ \begin{align} 36100= 114^2+152^2=6^2|10^2\\36121 =20^2+ 189^2=6^2|11^2\\36121 =61^2+ 180^2=6^2|11^2\\36196=40^2+ 186^2=6^2|14^2\\36324 =90^2+ 168^2=6^2|18^2\\36361 =60^2 +181^2=6^2|19^2\\36361=125^2+ 144^2=6^2|19^2\\36441=96^2+ 165^2=6^2|21^2\\36529=48^2+ 185^2=6^2|23^2\\36625=12^2+ 191^2=6^2|25^2\\36625=56^2+ 183^2=6^2|25^2\\36625=65^2+ 180^2=6^2|25^2\\36625=105^2+ 160^2=6^2|25^2\\36676=24^2+190^2=6^2|26^2\\36676=80^2+174^2=6^2|26^2\\36900 =6^2+ 192^2=6^2|30^2\\36900=48^2+ 186^2=6^2|30^2\\36900= 120^2+ 150^2=6^2|30^2 \end{align} $$The first of these numbers (36100) corresponds to Monday, February 3rd, 2048 by which time I'll be almost 88. Maybe I'll make it, maybe I won't.

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:
$$ 

\begin{align} \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) \end{align}
$$Hence \(n\) is always divisible by the difference. Take \(n=26975\) as an example. $$ \begin{align} \frac{27000}{25} &=1080\\26975=25 \times (1080-1) &=25 \times 1079 \end{align}$$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

Cubic Numbers

Today, January 11th 2016, I'm 24389 days old and what's special is that this number is 29 cubed or 29 x 29 x 29. Days like this are rare. For example, \(28^3 \) or 21592 occurred on May 10th 2009 and \(30^3 \) or 27000 will occur on March 6th 2023. Cubes of prime numbers are even rarer of course. The prime preceding 29 is 23 and \(23^3 \) or 12167 occurred on July 26th 1982. The prime following 29 is 31 and \( 31^3 \) or 29791 will occur on October 26th 2030 when I'm 81 years of age (if I make it that far). 

24389 has a surprisingly large number of entries in the Online Encyclopaedia of Integer Sequences (OEIS), 174 in fact which is unusual for a composite number of this magnitude. The first entry is for OEIS A000578: the cubes \( a(n) = n^3 \). The sequence, up to 24389 when n=29, looks like this:

0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 13824, 15625, 17576, 19683, 21952, 24389

The next entry is OEIS A030078: cubes of primes. The sequence, up to 24389, is: 8, 27, 125, 343, 1331, 2197, 4913, 6859, 12167, 24389.

From WolframAlpha, we find that 24389 is also a cube that is expressible as the sum of two squares in two different ways: 

\(24389 = 58^2+145^2  = 65^2+142^2 \)

Additionally, we find that 24389 is the hypotenuse of a primitive Pythagorean triple: 
  \(24389^2 = 15939^2+18460^2 \). So, all in all, an interesting number.