As with the EVEN + and ODD - and EVEN - and ODD + recursive algorithms, I've taken the original 393 page data document and converted it to a 83 page PDF that is more reader friendly. Here some excerpts.
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I came across this interesting snippet of information today:
The trick works because the conversion slope between Celsius and Fahrenheit:$$\frac{9}{5} = 1.8$$closely mirrors how reversing a two-digit number changes its numerical value in base-10 arithmetic, combined with simple integer rounding.
The standard conversion formula from Celsius ($C$) to Fahrenheit ($F$) is:
Every time $C$ increases by 12°C, $F$ increases by:
which is approximately 22°F.
Evaluating each step from the image reveals how closely the exact arithmetic matches the digit reversal:
Let a two-digit Celsius value be represented by digits $a$ and $b$:
Its digit reversal $R(C)$ is:
If you add $12$ to $C$, the tens digit increases by $1$ and the units digit increases by $2$ (assuming no carryover):
The reversal of this new value is:
Because reversing digits after adding $12$ increases the reversed value by $21$, and the physical temperature conversion rate is $1.8 \times 12 = 21.6$, the rounded whole-number sequences track together with remarkable accuracy across the everyday range from 0°C to 50°C.
The number associated with my diurnal age today (28262) is what might be termed a "super sphenic number" as I'll explain in this post. Firstly however, its factorisation:$$28262=2 \times 13 \times 1087$$If we reverse its digits, we get the number 26282 and this number is also sphenic:$$26282 = 2 \times 17 \times 773$$Let's now concatenate the factors of 28262 in ascending order. This gives us the number 2131087 which is also sphenic:$$2131087 = 7 \times 167 \times 1823$$28262 has a sum of digits of 20 and if we add this to the original number we get the palindromic number 28282 which is also sphenic:$$28262+20=28282 = 2 \times 79 \times 179$$The number has a product of digits of 384I and if we subtract this from the original number we get 27878 which is sphenic:$$28262 - 384 = 27878 = 2 \times 53 \times 263$$If we consider only the internal digits of 28262, we get the number 826 which is also sphenic:$$2\, 826 \, 2 \rightarrow826=2 \times 7 \times 59$$When a sphenic number is considered as a sphenic brick then it has an associated number in the form of the brick's surface area. In the case of 28262, this associated surface area is 32662 square units and this number too is sphenic:$$32662 = 2 \times 7 \times 2333$$28262 has a sum of proper divisors that is also sphenic:$$ \sigma(28262) - 28262 =17434 = 2 \times 23 \times 379 $$The number has a sum of prime factors (1102) that is sphenic:$$2 +13+1087=1102 = 2 \times 19 \times 29$$28262 has a totient of 13032 which is not sphenic but its cototient (number - totient) of 15230 is:$$28262 - 13032 = 15230 = 2 \times 5 \times 1523$$So we can see that 28262 may well be termed a super sphenic number because of the above associations.
Some time ago I got Gemini to create a 197 page document that identifies attractors and vortices arising from the recursive ODD - and EVEN + algorithm. It also lists their number of captives. Today I got Gemini to summarise and reformat this information so that it is more readable using the exact same template it created for my August 15th post titled ODD + and EVEN - Improved Format. That 142 page document can be located here. Here are some excerpts:
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Sometimes it's difficult to find many interesting properties for numbers greater than 28000. However, 28260 is definitely not one of those sorts of numbers. It has a plethora of interesting properties and in this post I'll list some of them. Firstly though, let's list its prime factorisation:$$ \textbf{28260} = 2^2 \times 3^2 \times 5 \times 157$$PROPERTY 1: sum of two squares
As can be seen, the number is a product of a power of 2 (\( 2^2\)), a 4\(k\)+1 prime raised to an even power (\(3^2\)) and two 4\(k\)+1 primes (5 and 157). This means that it can be expressed as a sum of two squares in two different ways, viz.:$$ \begin{align} \textbf{28260} &= 6^2+168^2\\ \textbf{28260} &= 96^2+138^2 \end{align} $$PROPERTY 2: a, b, c, d number
It is what I've termed an \(a, b, c, d\) number because its digits can be rearranged to form three different numbers with the property that \(a+b+c=d\) where all four numbers share identical digits. In this case, there are two possible arrangements:$$ \begin{align} 26082 + 26280 + \textbf{28260} &= 80622\\ 26280 + 28062 + \textbf{28260} &= 82602 \end{align} $$PROPERTY 3: d-powerful number
It is digitally powerful (or \(d\)-powerful) because it can be expressed as a sum of positive powers of its digits. Here we have:$$ \textbf{28260} = 2^{14} + 8^4 + 2^2 + 6^5 + 0$$Of additional interest is that this number is the beginning of chain of ten consecutive numbers with this property. See Figure 1.
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It a gapful number defined as a number of at least 3 digits that is divisible by the number formed by its first and last digits. Here the first and last digits form the number 20 and 20 is indeed a divisor:$$ \frac{ \textbf{28260}}{20} = 1413$$PROPERTY 6: untouchable number
It is an untouchable number defined as a number \(n\) that is not the sum of the proper divisors of any number \(k\). In other words:$$ \textbf{28260} \neq \sigma(k)-k \text{ for any }k $$PROPERTY 7: inconsummate number
It is an inconsummate number defined as a number \(n\) for which there is no number $k$ such that \(k\) divided by its sum of digits (SOD) gives $n$. Thus we have:$$ \frac{k}{\text{SOD}(k)}\neq \textbf{28260} \text{ for any } k$$PROPERTY 8: tau number
is a tau number since it is divisible by its number of divisors. Here there are 36 divisors and we have:$$ \frac{\textbf{28260}}{36}=785$$PROPERTY 9: Harshad number
It is a Harshad number defined as number that is divisible by the sum of its digits. Here we have a sum of digits of 18 and:$$ \frac{\textbf{28260}}{18}=1570$$PROPERTY 10: zeroes of the Mertens function
It is a number where the Mertens function has a value of zero. It forms a pair of consecutive numbers with 28259.
I examined the Möbius and Mertens functions in a post titled The Möbius Function and Mertens Function on January 25th 2020. In number theory, we define the Mertens function as:$$M(n) = \sum_{1\le k \le n} \mu(k)$$where \( \mu (k)\) is the Möbius function. For any positive integer n, \(μ(n)\) has values in {−1, 0, 1} depending on the factorisation of \(n\) into prime factors:$$\mu(n) = \begin{cases} 1 & \quad \text{if } n \text{ is square-free + integer with even number of prime factors}\\ -1 & \quad \text{if } n \text{ is square-free + integer with odd number of prime factors}\\ 0 & \quad \text{if } n \text{ has a squared prime factor} \end{cases}$$Figure 2 shows the situation:
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Some time ago I got Gemini to carry out an analysis for me of attractors and vortices in the range up to 40000 under the ODD + and EVEN - algorithm. I copied the output into a Google document that listed:
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Let's revisit the ODD + and EVEN - algorithm that I first discussed in a post titled Odds and Evens from June of 2021. In that post, I looked at the trajectory lengths of numbers up to 100,000 and Figure 1 shows a graph summarising what I found.
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Figure 1: permalink |
A036301 | Numbers whose sum of even digits and sum of odd digits are equal. |
I then extended the range to 200,000 and found a number (158893) that required 91 steps before it entered a loop or, to put it another way, it was captured by a vortex. In this case, the vortex consisted of the vorticals 160028, 160013, 160012, 160006, 159995, 160033, 160034. Figure 2 shows a graph of the trajectories:
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Figure 2: permalink |
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Figure 3: permalink (will need a Jupyter notebook) |
