Today I turned 26366 days old and yesterday, of course, I was 26365 days old. Each of these numbers have the property that the leading two digits (26) represent the number of fortnights in a year while the final three digits (365 and 366) represents the number of days in non-leap years and a leap years respectively. Incidentally, 26 fortnights represent 26 x 14 = 364 days and so 26364 encodes that fact. Similar pairs of numbers include:
- 12365 and 12366 where 12 represents the number of months in a calendar year
- 13365 and 13366 where 13 represents the number of lunar months in a calendar year
- 52365 and 52366 where 52 represents the number of full weeks in a calendar year
The concatenation of numbers, according to WolframMathWorld, is represented by the symbol \( \parallel \) and thus 26\( \parallel \)366 = 26366. The formula for the concatenation in base \(b\) of two numbers, \(p\) and \(q\), is given by:$$p \parallel q=p \, b^{f(q)} + q \text{ where } f(q)=\left \lfloor \log_b{q} \right \rfloor + 1$$$$ \text{where }f(q) \text{ represents the length of } q$$Let's test this formula out in the case of 26366. We have \(p=26\) and \(q=366\). Clearly the length of \(q\) is 3 but let's check using \( \left \lfloor \log_{10}366 \right \rfloor + 1\). The value of this is indeed 3 so all is good.
When constructing an algorithm, the formula is useful as a way of accomplishing the concatenation without resorting to strings. Figure 1 shows the relevant SageMath code.
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Figure 1: permalink |
26365 and 26366 have some interesting shared properties. They are both semiprimes and emirpimes :
- 26365 = 5 * 5273 and 56362 = 2 * 28181
- 26366 = 2 * 13183 and 66362 = 2 * 33181
In recreational mathematics, there are some interesting applications of concatenation. One of these involves so-called home primes. These are primes obtained by repeatedly factoring the increasing concatenation of prime factors of a given number. I've written about these in an eponymous post on May 2nd 2021. Let's work out the home primes for 26364, 26365 and 26366.
For 26364, we find ten steps are required to reach the home prime:
2636422313131379282444710113956473211479334437135897394094059117579999313530391933331113305413717887313175353597790561For 26365, only six steps are required:2636555273311783734271331178731764937For 26366, seven steps are required:2636621318333236871719551135731837196728069365576023
The Smarandache–Wellin numbers involve the concatenations of the first prime numbers. These numbers form OEIS A019518:
A019518 | Smarandache-Wellin numbers: a(n) is the concatenation of first n primes (written in base 10). |
The sequence begins:
223235235723571123571113235711131723571113171923571113171923235711131719232923571113171923293123571113171923293137235711131719232931374123571113171923293137414323571113171923293137414347
Of the numbers listed here only 2, 23 and 2357 are prime. The next such number that is prime has 355 digits!
If we concatenate the integers, we create the Champernowne constant, a transcendental real constant whose decimal expansion has important properties. It is named after economist and mathematician D. G. Champernowne, who published it as an undergraduate in 1933. I've written about this constant in an eponymous post on March 22nd 2019. The constant begins 12345678910111213141516 ...
The Copeland–Erdős constant is formed by the concatenation of "0." with the base 10 representations of the prime numbers in order. Its value, using the modern definition of prime, is approximately 0.235711131719232931374143…
There is a reverse integer sequence that comprises OEIS A000422:
A000422 | Concatenation of numbers from n down to 1. |
The sequence begins:
121321432154321654321765432187654321987654321109876543211110987654321121110987654321131211109876543211413121110987654321151413121110987654321161514131211109876543211716151413121110987654321181716151413121110987654321
Interestingly, this sequence produces very few primes with the first being when \(n=82\) and the next being \(n=37765\).
We can concatenate the odd, even, triangular, square, cubic and Fibonacci numbers and all of these have associated OEIS sequences. Follow this link for more information.
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