A Prolific Pair: 364585 and 820359

What do the numbers 364585 and 820359 have in common? Well, they're both odd for starters. However, although it's not obvious, they are both prolific in the sense that when squared, they produce numbers that have seven sequential, identical, non-zero digits (permalink):$$ \begin{align} 364585^2 &= 132922222225\\820359^2 &= 672988888881 \end{align}$$They are the only numbers with this property in the range up to one million. The first number mentioned (364585) makes an appearance in OEIS  A167712 for the case of \(n=7\):

A167712

\(a(n) \) = the smallest positive number, not ending in 0, whose square has a substring of exactly \(n\) identical digits.

The sequence begins: 1, 12, 38, 1291, 10541, 57735, 364585, 1197219, 50820359, 169640142, 298142397, 4472135955, 1490711985, 2185812841434

This same number makes another appearance in OEIS A048347 for the case of \(n=8\) where it should be noted that the identical digits are not necessarily sequential

A048347

\(a(n)^2\) is the smallest square containing exactly \(n\) 2's.

The sequence begins: 5, 15, 149, 1415, 4585, 14585, 105935, 364585, 3496101, 4714045, 34964585, 149305935, 1490725415, 4714469665, 1490711985, 149071333335, 1105537083332, 1489973900149, 15106363633335, 47140462469223

If we look at the cubes of numbers, then not surprisingly more numbers satisfy, specifically (permalink):$$ \begin{align} 339247^3 &= 39043437522222223 \\475741^3 &= 107674222222294021 \\ 720993^3 &= 374794444444986657 \\ 822389^3 &= 556201144444449869 \\ 699637^3 &= 342466666667067853 \\ 360598^3 &= 46888888826167192 \\ 948083^3 &= 852195188888887787 \\985426^3 &= 956912108888888776 \\ 764149^3 &= 446204706999999949 \end{align} $$So nothing profound in this post, just some peculiarities of the base 10 number system that produce runs of seven identical non-zero digits for certain squares and cubes of numbers up to one million.

Partially Pandigital Numbers

I've written about pandigital numbers before, specifically in posts named and dated:

• Pronic Pandigital Numbers and Beyond on July 23rd 2021
• Pandigital Numbers Formed From Squares on July 23rd 2021
• Pandigital Numbers Formed From the Product of a Number and its Reversal on July 9th 2018

An example of a number that forms a palindrome when multiplied by its reversal is 25299 x 99252 = 2510976348 where the digits from 0 to 9 appear and each only once. I was stimulated to propose an extension or modification of this idea of "pandigitalism" when confronted with finding some interesting properties of the number 26571.

It seems to be a number sorely lacking in interesting properties until I was reminded when consulting Number Academy that when this number is doubled it contains the digits from 1 to 5 with each digit occurring only once. In other words 26571 x 2 = 53142. I've chosen to refer to such an occurrence as "partially pandigital" and this property of 26571 can be generalised as follows:  

S040: Five digit numbers (not including leading zeros) that, when doubled, contain only the five digits 1, 2, 3, 4 and 5 in any order.  (link)

There are 36 members of this sequence and they are:

10677, 10767, 11577, 11757, 12567, 12657, 15627, 15726, 15762, 15771, 16077, 16257, 17076, 17256, 17562, 17571, 17607, 17706, 20676, 20766, 21576, 21756, 22566, 22656, 25617, 25662, 25671, 25716, 26067, 26157, 26562, 26571, 26607, 26706, 27066, 27156

Unlike squaring or multiplication by the reversal, these numbers arise somewhat simply by merely doubling. All the numbers have a digital root of 3 which is to be expected because the digital root of a number containing the digits 1, 2, 3, 4 and 5 is 6.

Of course, the five digits don't need to be sequential but things are more interesting when they are. Let's consider the digits 5, 6, 7, 8 and 9.

S041: Five digit numbers (not including leading zeros) that, when doubled, contain only the five digits 5, 6, 7, 8 and 9 in any order. (link)

There are 48 members of this sequence and they are:

28399, 28489, 28849, 28948, 28984, 28993, 29398, 29488, 29839, 29884, 29893, 29938, 32899, 32989, 33799, 33979, 34789, 34879, 37849, 37948, 37984, 37993, 38299, 38479, 39298, 39478, 39784, 39793, 39829, 39928, 42898, 42988, 43798, 43978, 44788, 44878, 47839, 47884, 47893, 47938, 48289, 48379, 48784, 48793, 48829, 48928, 49288, 49378

An example is 28399 that when doubled gives 56798. All the numbers have a digital root of 4 which is to be expected because the digital root of a number containing the digits 5, 6, 7, 8 and 9 is 8.

I've developed a flexible algorithm (permalink) in SageMath that allows the number of multiples to be varied along with the matching digits. Here is an example using a multiple of 4 and matching digits 0, 1, 2, 3 and 4:

L=[]

multiple=4

S=[0,1,2,3,4]

for n in [10000..int(100000/multiple)]:

d=n*multiple

if sorted(d.digits())==S:

L.append(n)

print("There are",len(L),"such numbers. They are:")

print(L)

for n in L:

print(n,"-->",n*multiple)

There are 6 such numbers. They are:

[10033, 10078, 10258, 10330, 10753, 10780]

10033 --> 40132

10078 --> 40312

10258 --> 41032

10330 --> 41320

10753 --> 43012

10780 --> 43120

Here we see for example that 10033 x 4 = 40132.

Once again, we see that seemingly uninteresting numbers do have interesting properties awaiting discovery if we are persistent enough in our investigation. I've written on this theme only recently in my post titled Unremarkable Numbers on November 25th 2021 and in AD and BC Numbers on December 3rd 2021.