The Cantor Ternary Set Revisited

It was back in January 2nd of 2022 (almost four years ago now) that I posted about The Cantor Ternary Set. I began the post with the following statement:

Today I turned 26572 days old and discovered that \( \dfrac{1}{26572} \) is in the Cantor set.

Infographic created using NotebookLM

In the context of the Cantor ternary set, 26572 is referred to as a \( \textbf{primitive} \) number and as such belongs to OEIS A173793. These numbers are not multiples of 3 as some of the other numbers are that belong in this set. The next such number is \( \textbf{28009} \), my diurnal age today. The initial members of this sequence are:

1, 4, 10, 13, 28, 40, 82, 91, 121, 244, 328, 364, 730, 757, 820, 949, 1036, 1093, 2188, 2362, 2812, 2920, 3280, 6562, 6643, 7381, 9490, 9841, 19684, 20440, 26248, 26572, 28009, 29524, 59050, 59293, 63973, 65620, 66124, 66430, 84253, 88573, 177148

As can be seen, after 28009, there is only one other number that I'm likely to experience in my lifetime and that is 29524. However if we include numbers in the Cantor set that are multiples of 3 then there are many more terms and these constitute OEIS A121153. Between 26572 and 29524, we have the following numbers as shown in Table 1 where the numbers (primitive and non-primitive) are shown together with their factorisations and the representation of their reciprocals in ternary format:

Table 1: permalink

For the non-primitive numbers, the factor of 3 is recurrent whereas in the primitive numbers it is absent. Here is a link to a video (uploaded to YouTube) that I got NotebookLM to create for me about the Cantor ternary set and below is the video:

Palindromic Day 27972

Today is palindromic day 27972 and the last palindromic day of the current millenium. The next palindromic day will occur in the new millenium and will 28082, one hundred and ten days from now.

My first observation is that the digits on either side of the central 9 also add to 9 to give a 9 - 9 - 9 pattern with 9 being the arithmetical digital root as well. This is the first and last time that such a triple pattern will occur in the current millenium. It cannot occur in the next two millenia (2 8 x 8 2 and 2 9 x 9 2) because the digits on either side of the central digit add to 10 and 11 respectively. So already 27972 is rather special. 

27972 is also a member of OEIS A344422: palindromes having more divisors than all smaller palindromes. The table below lists the initial members of this sequence and it can be seen that 27972 has a record 48 divisors.

  number   factorisation        divisors

  1        1                    1

  2        2                    2

  4        2^2                  3

  6        2 * 3                4

  44       2^2 * 11             6

  66       2 * 3 * 11           8

  252      2^2 * 3^2 * 7        18

  2112     2^6 * 3 * 11         28

  2772     2^2 * 3^2 * 7 * 11   36

  6336     2^6 * 3^2 * 11       42

  27972    2^2 * 3^3 * 7 * 37   48

  48384    2^8 * 3^3 * 7        72

27972 is also a member of OEIS A020485: least positive palindromic multiple of \(n\), or 0 if none exists. Here are the multiples for the initial values of \(n\).

  count   n       multiple

  1       1       1

  2       2       1

  3       3       1

  4       4       1

  5       5       1

  6       6       1

  7       7       1

  8       8       1

  9       9       1

  10      0       0

  11      11      1

  12      252     21

  13      494     38

  14      252     18

  15      525     35

  16      272     17

  17      272     16

  18      252     14

  19      171     9

  20      0       0

  21      252     12

  22      22      1

  23      161     7

  24      696     29

  25      525     21

  26      494     19

  27      999     37

  28      252     9

  29      232     8

  30      0       0

  31      434     14

  32      2112    66

  33      33      1

  34      272     8

  35      525     15

  36      252     7

  37      111     3

  38      494     13

  39      585     15

  40      0       0

  41      656     16

  42      252     6

  43      989     23

  44      44      1

  45      585     13

  46      414     9

  47      141     3

  48      2112    44

  49      343     7

  50      0       0

  51      969     19

  52      676     13

  53      212     4

  54      27972   518

  55      55      1

  56      616     11

  57      171     3

  58      232     4

  59      767     13

  60      0       0

  61      26962   442

27972 requires 13 steps to reach the palindrome 4964444694 under the Reverse and Add algorithm. 

27972 is a decagonal or ten-sided number and is the 84th decagonal number and the second non-trivial palindromic decagonal number after 232. See Figure 1 where 232 is shown but not 27972.

Figure 1: source

27972 is a member of OEIS A356854: palindromes that can be written in more than one way as the sum of two distinct palindromic primes. In the case of 27972 we have:

10501 + 17471 = 27972

11311 + 16661 = 27972

11411 + 16561 = 27972

12421 + 15551 = 27972

Another Weird Day

It's been 560 days since my last \( \textbf{weird} \) day. Today I turned \( \textbf{27790} \) days old and once again it's weird. I've written about this type of number before in blog posts titled Weird Numbers on August 4th 2019 and also in an earlier post titled Zumkellar Numbers, Half Zumkellar Numbers and Pseudoperfect Numbers on November 22nd 2018.

Most abundant numbers are pseudoperfect meaning that a subset of their proper divisors can be chosen so that their sum is equal to the number. If the sum of all the proper divisors of a number equals the number itself, then the number is said to be perfect. The first few perfect numbers are 6, 28, 496, 8128 and 33550336. An example of an abundant number that is pseudoperfect is 24 with proper divisors of 1, 2, 3, 4, 6, 8 and 12. Adding these gives 36 so the number is clearly abundant. However, if we add the proper divisors without including 12, we reach 24 and so the number is pseudoperfect.

There are two types of weird numbers: primitive and non-primitive. A non-primitive weird number is a number that is a multiple of a weird number. The primitive weird numbers up to 40000 are 70, 836, 4030, 5830, 7192, 7912, 9272, 10792 and 17272. Here is a list of all weird numbers (primitive and non-primitive) up to 40000:

70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, 15610, 15890, 16030, 16310, 16730, 16870, 17272, 17570, 17990, 18410, 18830, 18970, 19390, 19670, 19810, 20510, 21490, 21770, 21910, 22190, 23170, 23590, 24290, 24430, 24710, 25130, 25690, 26110, 26530, 26810, 27230, 27790, 28070, 28630, 29330, 29470, 30170, 30310, 30730, 31010, 31430, 31990, 32270, 32410, 32690, 33530, 34090, 34370, 34930, 35210, 35630, 36470, 36610, 37870, 38290, 38990, 39410, 39830, 39970

All the numbers above with the exception of the primitive weird numbers in red are multiples of 70. Any weird number multiplied by a prime number that is greater than the sum of the divisors of the numbers is itself weird. Thus 70 has a sum of divisors of 144 and the first prime above this is 149. 70 x 149 produces the weird number 10430.

As can be seen, my next weird number (28070) is 280 days away and after that there is a gap of 560 days to my next weird number (28630). The gaps of course are determined by the distance between successive prime numbers. Thus:$$ \begin{align} 27790 &=70 \times 397\\28070 &=70 \times 401 \\28630 &=70 \times 409 \end{align} $$Here is a fuller list of primitive weird numbers (link):

70, 836, 4030, 5830, 7192, 7912, 9272, 10792, 17272, 45356, 73616, 83312, 91388, 113072, 243892, 254012, 338572, 343876, 388076, 519712, 539744, 555616, 682592, 786208, 1188256, 1229152, 1713592, 1901728, 2081824, 2189024, 3963968, 4128448

While this post is a little repetitive, I feel it's important to acknowledge the occurrence of weird numbers when they occur because of their rarity.

An Interesting Prime

Being born on the 3rd April 1949, my date of birth is often represented as 3 - 4 - 49. These numbers when concatenated form the prime number 3449. I was reminded of this number because of the factorisation of the number associated with my diurnal age today, 27592.$$27592 = 2^3 \times 3449 = 8 \times 3449$$So today my life can be divided into exactly eight equal parts, each of them 3449 days long which is about 9.44 years. The previous multiple$$7 \times 3449 = 24143$$occurred on May 10th 2015 when I was still working at the Shanghai Singapore International School. The next multiple$$9 \times 3449 = 31041$$ will fall on March 29th 2034, shortly before my 84th birthday (if I make it that far).

3449 forms the initial prime of a Cunningham chain of the first type with length exactly 3 and so: $$ 3449 \rightarrow 2 \times 3449 + 1 = 6889 \text{ (prime)} \\ 6889 \rightarrow 2 \times 6889 +1 = 13799 \text{ (prime)}$$Primes with this property form OEIS A059762. Another prime-related property of 3449 qualifies it for membership in OEIS A088483:

A088483: primes \( \textit{p} \) such that \(p^2+p-1\) and \(p^2+p+1\) are twin primes.

For 3449, the twin primes are \(11899049\) and \(11899051\). 

3449 is also a home prime with a homeliness of 3 because:$$ \begin{align} 611 &= 13 \times 47 \rightarrow 1347\\ 1347 &= 3 \times 449 \rightarrow 3449 \end{align}$$Not all primes are home primes of course. Take 613 as an example of a prime that is not a home prime because it cannot be formed by the concatenation of the prime factors of any number (the prime factors need to be concatenated in ascending order).

3449 is also a member of OEIS A153116:

A153116: primes \(p\) such that \(p^2 +12\) and \(p^2-12\) are also primes.

Here the two primes are \(11895589\) and \(11895613\). Additionally:$$ \text{period of}\frac{1}{3449}=\frac{3449-1}{8} = 431$$This property qualifies 3449 for membership of OEIS A056213:

A056213: primes \(p\) for which the period of reciprocal = \(\dfrac{p-1}{8}\).

3449 is a Sophie Germain prime because:$$2 \times 3449+1=6899 \text{ is prime}$$3449 also features in so-called "Golden Semiprimes" and this qualifies it for membership in OEIS A108544:

A108544: primes that are factors of distinct golden semiprimes (A108540).

I posted about these types of semiprimes in Semiprime Factor Ratios way back on the 26th August 2016. In that post I said that:

A golden semiprime is a number that factors to:

• \(p \times q\) where \(p\) and \(q\) are prime
• \( | \,p \, \phi - q \,| <1\) where \(\phi=\dfrac{\sqrt{5}+1}{2} \)

In the case of 3449, it is the \(q\) and \(p=2131\) and the golden semiprime is:$$7349819=2131 \times 3449$$The OEIS mentions 314 sequences in which 3449 makes an appearance and I've only dealt with a few of them here. However, I see 3449 as an important number in my life and didn't want its current occurrence to pass unnoticed.

A Special Fourth Power Diophantine Equation

Consider the Diophantine equation:$$a^4+b^4+c^4+d^4=(a+b+c+d)^4$$It's clear that no set of positive numbers will satisfy this equation, at least one of the numbers will need to be negative. It's also clear that if we find a solution \(n=a+b+c+d\) then this will generate an infinity of solutions between all multiples of this number will satisfy the equation:$$ \begin{align} (na)^4+(nb)^4+(nc)^4+(nd)^4 &= (n *(a+b+c+d))^4\\n^4  a^4 +n^4  b^4 + n^4  c^4 + n^4  d^4 &= n^4  (a+b+c+d)^4 \\a^4+b^4+c^4+d^4 &= (a+b+c+d)^4 \end{align} $$where \(n\) can be any integer, positive or negative, or even zero. So what is the first number that satisfies the equation? In 1964, somebody named Brudno found the number 5491 that satisfies where  5491 is written as 955 + 1700 - 2364 + 5400 and we have:$$955^4+1700^4+(-2364)^4+5400^4\\=(955 + 1700 - 2364 + 5400)^4$$In terms of positive numbers, this means that the progressive multiples of 5491 are also solutions viz. 10982, 16473, 21964, 27455, 32946, 38437, 43928, 49419, ... etc..

Are there other numbers? Well somebody named Wroblewski found 51361 to be a solution when it written as 48150 - 31764 + 27385 + 7590. Thus:$$48150^4 +(-31764)^4+27385^$+7590^4\\=(48150 - 31764 + 27385 + 7590)^4$$At this point, it's relevant to include this quote from the comments to OEIS A138760: numbers \(n\) such that \(n^4\) is a sum of 4th powers of four nonzero integers whose sum is \(n\):

Any multiple of a member is also a member. A member that is not a multiple of another member is called primitive. Using elliptic curves, Jacobi and Madden prove that there are infinitely many primitive members. According to them, the only primitive members less than 222,000 are 5491 (due to Brudno) and 51361 (due to Wroblewski).

The comments then list a number greater than 220,000 and that number is 1347505009. The reason I came across these numbers is that today my diurnal age is 27455 and that is the fifth multiple of 5491.

Some Properties of the Number 2023

For my one hundredth post of the year 2022, I thought I'd create a post about the forthcoming year before it arrives. Firstly, looking at Numbers Aplenty, we find that a 4×4 square can be tiled in 2023 ways using the L triomino and the monomino. Figure 1 shows four examples and click here to see all 2023 possible arrangements. See OEIS A220054 and A127870.

Figure 1

Another interesting property that Numbers Aplenty brings up is the following:

2023 = (2+0+2+3)(2²+0²+2²+3²)²

It also turns out that the prime factors, when concatenated, form a palindrome.

2023 = 7 x 17 x 17 --> 71717

This as it turns out is not a common property (see OEIS A046447). In the range of composite numbers up to 40,000, there are only 193 such numbers. In terms of years, it has been 320 years since this last happened and next occurrence will be rather sooner, in 2048. Here are the numbers (permalink):

4, 8, 9, 16, 25, 27, 32, 39, 49, 64, 69, 81, 119, 121, 125, 128, 129, 159, 219, 243, 249, 256, 259, 329, 339, 343, 403, 429, 469, 507, 512, 625, 669, 679, 729, 795, 1024, 1207, 1309, 1329, 1331, 1533, 1547, 1587, 1589, 1703, 2023, 2048, 2097, 2187, 2319, 2321, 2359, 2401, 2649, 2701, 3039, 3125, 3421, 3503, 3629, 3633, 3639, 3729, 3897, 3899, 3975, 4096, 4303, 4607, 4719, 4839, 5289, 5295, 5547, 5565, 5603, 5739, 6209, 6401, 6561, 6591, 6597, 6979, 7269, 7663, 7819, 7833, 8029, 8192, 8203, 8427, 8519, 8745, 8921, 9329, 9583, 9703, 9707, 9795, 10001, 10021, 10201, 10299, 10599, 10731, 11199, 11499, 12207, 12729, 12795, 13129, 13489, 13573, 13801, 13929, 14133, 14147, 14189, 14295, 14399, 14641, 14829, 15459, 15463, 15549, 15563, 15625, 15987, 16233, 16384, 16507, 16807, 16863, 16959, 17161, 17295, 17363, 17859, 18489, 18789, 18829, 19003, 19683, 19689, 19875, 19907, 20111, 20289, 20489, 20589, 20667, 21259, 21959, 22139, 22507, 22719, 22801, 23019, 23619, 24601, 24901, 25459, 26349, 27879, 28273, 28601, 28609, 28847, 28951, 29729, 29829, 31129, 31363, 31423, 31829, 32357, 32761, 32768, 33173, 33339, 34391, 35143, 36481, 36501, 36763, 36849, 38307, 38899, 38955, 39599, 39939, 39963

On the topic of palindromes, the sum of 2023 and its reverse 3202 is a palindrome. Of all the numbers from 1 to 40,000, there are 9456 (or almost 25%) with this property.

2023 + 3202 = 5225

2023 is also palindromic in base 16 since it equals 7e7. In the range of numbers up to 40,000, this occurs 409 times.

2023 = 7e7

The numbers are:

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 34, 51, 68, 85, 102, 119, 136, 153, 170, 187, 204, 221, 238, 255, 257, 273, 289, 305, 321, 337, 353, 369, 385, 401, 417, 433, 449, 465, 481, 497, 514, 530, 546, 562, 578, 594, 610, 626, 642, 658, 674, 690, 706, 722, 738, 754, 771, 787, 803, 819, 835, 851, 867, 883, 899, 915, 931, 947, 963, 979, 995, 1011, 1028, 1044, 1060, 1076, 1092, 1108, 1124, 1140, 1156, 1172, 1188, 1204, 1220, 1236, 1252, 1268, 1285, 1301, 1317, 1333, 1349, 1365, 1381, 1397, 1413, 1429, 1445, 1461, 1477, 1493, 1509, 1525, 1542, 1558, 1574, 1590, 1606, 1622, 1638, 1654, 1670, 1686, 1702, 1718, 1734, 1750, 1766, 1782, 1799, 1815, 1831, 1847, 1863, 1879, 1895, 1911, 1927, 1943, 1959, 1975, 1991, 2007, 2023, 2039, 2056, 2072, 2088, 2104, 2120, 2136, 2152, 2168, 2184, 2200, 2216, 2232, 2248, 2264, 2280, 2296, 2313, 2329, 2345, 2361, 2377, 2393, 2409, 2425, 2441, 2457, 2473, 2489, 2505, 2521, 2537, 2553, 2570, 2586, 2602, 2618, 2634, 2650, 2666, 2682, 2698, 2714, 2730, 2746, 2762, 2778, 2794, 2810, 2827, 2843, 2859, 2875, 2891, 2907, 2923, 2939, 2955, 2971, 2987, 3003, 3019, 3035, 3051, 3067, 3084, 3100, 3116, 3132, 3148, 3164, 3180, 3196, 3212, 3228, 3244, 3260, 3276, 3292, 3308, 3324, 3341, 3357, 3373, 3389, 3405, 3421, 3437, 3453, 3469, 3485, 3501, 3517, 3533, 3549, 3565, 3581, 3598, 3614, 3630, 3646, 3662, 3678, 3694, 3710, 3726, 3742, 3758, 3774, 3790, 3806, 3822, 3838, 3855, 3871, 3887, 3903, 3919, 3935, 3951, 3967, 3983, 3999, 4015, 4031, 4047, 4063, 4079, 4095, 4097, 4369, 4641, 4913, 5185, 5457, 5729, 6001, 6273, 6545, 6817, 7089, 7361, 7633, 7905, 8177, 8194, 8466, 8738, 9010, 9282, 9554, 9826, 10098, 10370, 10642, 10914, 11186, 11458, 11730, 12002, 12274, 12291, 12563, 12835, 13107, 13379, 13651, 13923, 14195, 14467, 14739, 15011, 15283, 15555, 15827, 16099, 16371, 16388, 16660, 16932, 17204, 17476, 17748, 18020, 18292, 18564, 18836, 19108, 19380, 19652, 19924, 20196, 20468, 20485, 20757, 21029, 21301, 21573, 21845, 22117, 22389, 22661, 22933, 23205, 23477, 23749, 24021, 24293, 24565, 24582, 24854, 25126, 25398, 25670, 25942, 26214, 26486, 26758, 27030, 27302, 27574, 27846, 28118, 28390, 28662, 28679, 28951, 29223, 29495, 29767, 30039, 30311, 30583, 30855, 31127, 31399, 31671, 31943, 32215, 32487, 32759, 32776, 33048, 33320, 33592, 33864, 34136, 34408, 34680, 34952, 35224, 35496, 35768, 36040, 36312, 36584, 36856, 36873, 37145, 37417, 37689, 37961, 38233, 38505, 38777, 39049, 39321, 39593, 39865

2023 is also a:

• Duffinian number: a composite number that has no factors in common with its sum of divisors \( \sigma(\tau)) \).
  
  
• Harshad number: a number that is divisible by the sum of its digits. In fact, 2022 was a Harshad number and so are 2024 and 2025 we have a run of four consecutive Harshad numbers.
  
  
• Lucky number: a number that survives a sieving process which is similar to the Eratosthenes sieve that can be used to compute the prime numbers.
  
  
• Junction number: a number that can be formed in two ways by adding a number to its sum of digits. Here we have 1997 + sod(1997) = 2015 + sod(2015) = 2023 where sod means sum of digits.

Turning to the Online Encyclopedia of Integer Sequences (OEIS), we find that it is a member of OEIS A063416:

A063416

Multiples of 7 whose sum of digits is equal to 7.

The sequence begins:

7, 70, 133, 322, 511, 700, 1015, 1141, 1204, 1330, 2023, 2212, 2401, 3031, 3220, 4102, 5110, 7000, 10024, 10150, 10213, 10402, 11032, 11221, 11410, 12040, 12103, 13111, 13300, 15001, 20041, 20104, 20230, 21112, 21301, 22120, 23002, 24010 

There is a strong connection between 2023 and the digit 7:

• the sum of its digits is 7
• it is divisible by 7
• both of its factors (7 and 17) contain the digit 7
• in octal, it has the representation 3747
• in hexadecimal, it has the representation 7e7
• in base 17, it has the representation 700
• its Fibonacci seed numbers are 17 and 49 = 7 x 7

By "Fibonacci seed numbers" I mean that by starting with 17 and 49 and proceeding Fibonacci-style, we get to 2023 thus:

17, 49, 66, 115, 181, 296, 477, 773, 1250, 2023

It is a member of OEIS A070258:

A070258

Smallest of 3 consecutive numbers each divisible by a square.

• \(2023 = 7 \times 17^2\)
• \(2024 = 2^3 \times 11 \times 23\)
• \(2025 = 3^4 \times 5^2\)

The initial members of the sequence are:

48, 98, 124, 242, 243, 342, 350, 423, 475, 548, 603, 724, 774, 844, 845, 846, 1024, 1250, 1274, 1323, 1375, 1420, 1448, 1519, 1664, 1674, 1680, 1681, 1682, 1848, 1862, 1924, 2007, 2023, 2056, 2106, 2150, 2223, 2275, 2348, 2366, 2523, 2527, 2574, 2644

Well, that's largely what I decided to present from Numbers Aplenty and the OEIS. What else is there of interest about the number 2023? I thought I'd ask ChatGPT the following question: what are some interesting facts, mathematical or otherwise, about the number 2023? Figure 2 is a screenshot of what it spat out.

Figure 2

As usual with ChatGPT, when it comes to Mathematics, it can spew out some nonsense. In case, it claims 2023 is a prime number! Similarly, it's claim that "2023 is the sum of three consecutive prime numbers: 661 + 673 + 677 is false. These are indeed consecutive prime numbers but they add up to 2011. Can it get any worse? Well, yes. The claim is made that "2023 is the hypotenuse of a Pythagorean triple: (683, 992, 2023)" and testing this out we find that \(683^2+992^2=1,450,553\) while \(2023^2=4,092,529\).  We're not done yet! Another bold claim is made that "2023 can be written as the sum of two cubes in two different ways: \(2023 = 13^3 + 2^3 = 11^3 + 10^3\) ". Nope. \(13^3+2^3=2205\) and \(11^3+10^3=2331\). It turns out that 2023 cannot be expressed as a sum of two cubes. A final claim is made that 2023 has the representation 11111011111 in binary. No. The correct representation is 11111100111.

Clearly time to give up on ChatGPT as a source of information about Mathematics. I've written about this AI bot's deficiencies in a post titled ChatGPT and a Tragi-Comedy of Integration. Let's not leave off without a final laugh. After the bot's spectacular failures, I asked it: 

I'm impressed in a negative way. Except for points 2 and 6, every mathematical statement that you made was  wrong. 2023 is not a prime number and so on. How can you get things so wrong? 

Its response was: 

I apologize for any errors in my previous response. As a language model, I can provide information and answer questions to the best of my ability based on the knowledge and information that I have been trained on. However, I am not infallible and it is possible for me to make mistakes or provide incorrect information.

In fact, 2023 is not a prime number. It can be divided evenly by 3, 11, and 61, so it is a composite number. I apologize for any confusion this may have caused.

I hope this clears up any misunderstandings and I apologize for any errors in my previous response. Please feel free to ask me any other questions you may have.

Actually, looking at the atomic number of Calcium, I checked and found that it was 20 not 2023. This perhaps gives an insight into what's going wrong with the bot. It's looking at the first two digits (20) and ignoring the last two digits (23). As for the Mathematics, I simply don't know how its coming up with its answers.

All triprimes can be envisaged as "bricks", 3-D solid objects. If all three primes are distinct, we have a sphenic number and an associated sphenic brick. The volume of the brick is the number itself but there is always an associated number: the surface area of the brick. In the case of 2023, the associated number is 1054 (square units). This number is sphenic and the associated sphenic brick has a surface area of 1246 square units. This number in turn is sphenic and the associated area is 1630 square units. Once again, 1630 is sphenic with associated surface area of 2302 square units. At this point, the chain is broken because 2302 is biprime. This is summarised below:$$ \begin{align} 2023 &= 7 \times 17 \times 17 &\rightarrow 1054\\1054 &= 2 \times 17 \times 31 &\rightarrow 1246\\1246 &= 2 \times 7 \times 89 &\rightarrow 1630\\1630&= 2 \times 5 \times 163 &\rightarrow 2302\\2302 &= 2 \times 1151 \end{align}$$Figure 3 shows a list of the triprime numbers between 2000 and 2100. Thus there is another twelve years before another triprime number, 2035, is encountered. 2035 is also a sphenic number because it has three distinct factors

Figure 3