What's Special About 45162?

I want to start a series of posts that highlight special numbers. Mostly I'll be drawing on material from earlier posts where a number was mentioned but perhaps not given the prominence that it deserved. Today's number is 45162 and it's unique in the number range from one to one million. Why?

\(45162\)

The reason is by no means obvious but it has to do with the catcatenation of its prime factors which are:

 \(2 \times 3^2 \times 13 \times 193\)

When we concatenate these prime factors, we get the following number:

\(23313193\)

This number happens to be a prime. In itself, this doesn't make 45162 very special, let alone unique. About 13.7% of numbers in the range up to one million have the property that, when their prime factors are concatenated in order from lowest to highest, the resulting concatenated number is prime. Such a prime is called the home prime of that number and thus 23313193 is the home prime of 45162.

What makes 45162 unique is that it is the start of a chain of eight consecutive numbers that all have this property as shown in Figure 1:

Figure 1

Quite remarkable and it's no wonder that this octet is the only one in the range up to one million. Interestingly, none of the home primes associated with this octet of numbers contains the digit 8. I first mentioned these numbers in a post titled One Step Away on January 31st 2022. 45162 is given prominence because it starts off the sequence.

Prime Weeks

As I number my days, I find that my prime days come and go but every Sunday, the day on which I was born, I know that it can never be a prime day because, whatever its number, it will always be divisible by 7. Today for example is a Sunday and I turned 25907 days old. Now 25907 = 7 x 3701 and so 3701 weeks have now elapsed since I was born. Of interest however, is the fact that 3701 is prime. There have been 517 Sundays where the number was biprime or semiprime.

𝓟𝓡𝓘𝓜𝓔 𝓦𝓔𝓔𝓚

I've chosen to call such weeks, beginning on Sunday, prime weeks. They can never occur consecutively except in the case of the second and third weeks. Each prime week will have its own character as determined by the nature of the larger prime. For example, what is significant about today's 3701?

𝓢𝓤𝓜 𝓞𝓕 𝓢𝓔𝓥𝓔𝓝 𝓟𝓡𝓘𝓜𝓔𝓢

One interesting property is that it is a prime that is itself the sum of seven consecutive primes. This property defines OEIS A082246 whose members, up to and including 3701, consist of the primes:

197, 223, 251, 281, 311, 401, 431, 463, 523, 593, 659, 719, 757, 827, 863, 947, 991, 1063, 1171, 1753, 1901, 2347, 2393, 2647, 2689, 2731, 2777, 2819, 2953, 3347, 3389, 3533, 3643, 3701

It's easy enough to generate this sequence from the following SageMath code:

P=Primes()
L=[]
for i in range(0,1000):
    sum=0
    for j in range(0,7):
        sum+=P[i+j]
    if is_prime(sum):
        L.append(sum)
    if sum==3701:
        break
print(L)

Click here for a SageMathCell permalink to the above code. Thus the themes of primes and sevens are echoed in this property of 3701.

𝓗𝓞𝓜𝓔 𝓟𝓡𝓘𝓜𝓔

3701 is also a member of OEIS A133960: home primes whose homeliness is 3. Without going into the definition of what is meant by a home prime and homeliness (here is the Wikipedia link), it can be seen that 206 = 2 x 103 --> 2103 = 3 x 701 --> 3701 and thus 3701 is a home prime of homeliness 3 because 206, 2103 and 3701 all lead to 3701.

The number 3701 attracts 203 entries in the OEIS so there are probably other properties that are of interest but that will suffice for now. Here is the SageMath code to generate the list of numbered Sundays, up to and including 25907, that start prime weeks:

L=[]
P=Primes()
number=0
for p in P:
        number=p*7
        L.append(number)
        if number==25907:
            break
print(L) 

Here is the permalink.

My next prime week will occur on day 25963 = 7 * 3709 which is 56 days or eight weeks away from today's 9th March 2020. The observance of such days provides, via the analysis of the number associated with that day, another opportunity to discover more about the properties of numbers. Since I began this blog on Wednesday, 30th September 2015, it's amazing how much I've learned and how much more there is to know. My very first post was about Catalan Numbers and was begun when I was 24285 days old.

At that time I knew nothing about SageMath or LaTeX. Even though I'd spent many years teaching Mathematics, my knowledge was largely confined to the topics that were included in the various high school curricula.

on Sunday May 2nd 2021