888 Revisited

In November of 2021, I posted about 888 from a mathematical and non-mathematical perspective. Mathematically, I noted that:

• 888 is the smallest cube in which each digit occurs exactly three times. The list up to one million of such numbers is (permalink):

888, 56592, 58524, 65577, 70869, 78183, 496941, 512427, 516267, 517461, 557949, 565920, 581421, 585558, 661959, 711828, 713772, 723627, 724983, 733053, 739563, 764472, 781830, 877242, 988458

• 888 is the only cube in which three digits occur three times. For example, the next number in the previous sequence (56592) has a cube of 181244621426688 but there are four digits that occur three times.
  
  
• 888 the smallest multiple of 24 whose digit sum is 24 and, as well as being divisible by its digit sum, it is divisible by all of its digits.
  
  
• 888 and 24 show up again in the former's membership of OEIS 236661 where 888 counts the number of partitions of 24 that have a standard deviation greater than 2. Permalink.
  
  
• Other properties of 888 include its being a happy, Harshad, Moran, nude, strobogrammatic, modest, congruent, amenable, practical, abundant, pseudoperfect and Zumkeller number (see Numbers Aplenty).
  
  
• The 8's are involved in 888 again thanks to its membership of OEIS A127335.
  
  

A127335

Numbers that are the sum of 8 successive primes.

 The sequence runs:

77, 98, 124, 150, 180, 210, 240, 270, 304, 340, 372, 408, 442, 474, 510, 546, 582, 620, 660, 696, 732, 768, 802, 846, 888

The eight successive primes in the case of 888 are 97, 101, 103, 107, 109, 113, 127 and 131 with an average of 111. Both 111 and 888 are of course repdigits along with the infamous 666 or number of the beast.

• 888 arises in the context of aliquot sequences via OEIS A014360:
  
  

A014360

Aliquot sequence starting at 552.

The sequence begins:

552, 888, 1392, 2328, 3552, 6024, 9096, 13704, 20616, 30984, ...

To quote from Wolfram Alpha:

It has not been proven that all aliquot sequences eventually terminate and become periodic. The smallest number whose fate is not known is 276. There are five such sequences less than 1000, namely 276, 552, 564, 660, and 966, sometimes called the "Lehmer five". 

In this post, I'll revisit 888 but I'll be doing so via 28192, the number associated with my diurnal age today. This number has the interesting property that its square and cube both contain the digit sequence 888:$$ \begin{align} 28192^2 &= 7947\textbf{888}64 \\ 28192^3 &= 22406687653\textbf{888} \end{align} $$There are only four such numbers in the range up to 40000 and they are 3192, 28192, 31920 and 33878. One of these, 31920, is simply a derivative of 3192. Of these four numbers, it is only 28192 that contains three eights when expressed in terms of its prime factors:$$2\textbf{8}192 = 2^5 \times \textbf{88}1$$It could even be written as \(2^2 \times \textbf{8} \times \textbf{88}1 \) so it is very giving in terms of its eightness!

What about other numbers in the range up to 40000 that both display three digit repdigits when squared and cubed? Here are the numbers that satisfy:

For repdigit \( \textbf{111}\), there is only one number that satisfies:

• \(10558^2= \textbf{111}471364 \text{ and } 10558^3 =117691466\textbf{111}2\)

For repdigits \( \textbf{222} \) and \( \textbf{333} \), no numbers satisfy but for repdigit \( \textbf{444}\) we have:

• \(6962^2 = 48469\textbf{444} \text{ and } 6962^3 = 337\textbf{444}269128\)
• \(12038^2 = 144913\textbf{444} \text{ and } 12038^3 = 17\textbf{444}68038872\)
• \(21081^2 = \textbf{4444}08561 \text{ and } 21081^3 = 936857687\textbf{444}1\)
• \(32538^2 = 1058721\textbf{444} \text{ and } 32538^3 = 3\textbf{444}8678344872\)
• \(37808^2 = 1429\textbf{444}864 \text{ and } 37808^3 = 540\textbf{444}51418112\)

For repdigit \( \textbf{555} \), again only one number satisfies:

• \(38152^2 =14\textbf{555}75104 \text{ and } 38152^3 = \textbf{555}33101367808\)

For the famous repdigit \( \textbf{666} \), we have two numbers that satisfy with some extra 6's thrown in for the case of \(30605\):

• \(17767^2 = 315\textbf{666}289 \text{ and } 17767^3 = 560844295\textbf{666}3\)
• \(30605^2 = 93\textbf{6666}025 \text{ and }30605^3 = 28\textbf{66666}3695125\)

For the repdigit \( \textbf{777} \), again two numbers satisfy:

• \( 18924^2 = 35811\textbf{777}6 \text{ and }18924^3= 6\textbf{777}020793024 \)
• \( 34753^2 =120\textbf{777}1009 \text{ and } 34753^3 = 41973665875\textbf{777} \)

For the repdigit \( \textbf{999} \) we have a bonanza so I'll just list the numbers and show one example:

\( 9997, 9998, 9999, 19998, 19999, 29999, 38729, 39999 \)

• \(9997^2 = \textbf{999}40009 \text{ and } 9997^3 = \textbf{999}100269973\)