SOD stands in a mathematical context for Sum of Digits and it can also be written as SoD or sod. Unlike a number's primeness or non-primeness, a number's SoD is peculiar to the number system being used and is thus of interest mainly in recreational mathematics. My diurnal age today is 26384 with a sum of digits of 23. When these two numbers are added together, the result is a prime number, 26407. This qualifies 26384 for inclusion in OEIS A047791:
A047791 | Numbers \(n\) such that \(n\) plus digit sum of \(n\) (A007953) equals a prime. |
- 26291 + sod(26291) = 26291 + 20 = 26311 and
- 26300 + sod(26300) = 26300 + 11 = 26311
Numbers represented by their own digits by certain operations are considered as selfie numbers. Some times they are called wild narcissistic numbers. There are many ways of representing selfie numbers. They can be represented in digit’s order, reverse order of digits, increasing and/or decreasing order of digits, etc. These can be obtained by use of basis operations along with factorial, squareroot, Fibonacci sequence, Triangular numbers, binomial coefficients, s-gonal values, centered polygonal numbers, etc. In this work, we have written selfie numbers by use of concatenation, along with factorial and square-root. The concatenation idea is used in a very simple way. The work is limited up to 5 digits. Work on higher digits shall be dealt elsewhere. Source.
I use the example of 25926 that can be expressed as:
((−2+5)!)!×C(9,2)+6 = (3!)! x 36 + 6 = 6! x 36 + 6 = 720 x 36 + 6 = 25926
Another example is \(39304:=((4||03)−9)^3\) where || stands for concatenation.
This is a big topic and it has been covered in detail in my prementioned blog post.
FRIEDMAN NUMBERS
These could be considered a subset of the selfie numbers but they form a category in their own right and are constructed much more simply. I wrote about these in a blog post from October 8th 2020 titled Forming Equations from Integers. To quote:
Consider \(28547=(8+5)^4−(7×2)\) expressed in base 10, both sides use the same digits. An integer is a Friedman number if it can be put into an equation such that both sides use the same digits but the right hand side has one or more basic arithmetic operators (addition, subtraction, multiplication, division, exponentiation) interspersed. Brackets, as usual, are essential to clarify the order of operations. These numbers are named after Erich Friedman, Assoc. Professor of Mathematics at Stetson University. With the help of his students he has researched Friedman numbers in bases 2 through 10 and even with Roman numerals. When both sides use the digits in the same order, the number is called a ”nice” or ”strong” Friedman number. For example, \(3125=(3+[1×2])^5\).
NARCISSISTIC NUMBERS
Selfie numbers are sometimes called wild narcissistic numbers but the proper narcissistic numbers. Numbers Aplenty defines them thus:
A number \(n\) of \(k\) digits is called narcissistic if it is equal to the sum of the \(k^{th}\) powers of its digits. For example, \(153\) is narcissistic because \(153 = 1^3+5^3+3^3\). Narcissistic numbers are also called Armstrong or plus-perfect numbers. It has verified that there in fact only 88 such numbers. Those up to one million are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474, 54748, 92727, 93084, 548834, 1741725, 4210818, 9800817, 9926315.
D-POWERFUL NUMBERS
D-powerful numbers are akin to narcissistic numbers but with more flexibility regarding the powers to which the digits may be raised. To quote from Numbers Aplenty:
An integer \(n\) is called digitally powerful (here d-powerful) if it can be expressed as a sum of positive powers of its digits. For example:$$3459872 = 3^1 + 4^6 + 5^5 + 9^6 + 8^3 + 7^7 + 2^{21}$$The first d-powerful numbers are:
1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 43, 63, 89, 132, 135, 153, 175, 209, 224, 226, 262, 264, 267, 283, 332, 333, 334, 357, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 407, 445, 463, 518, 598, 629, 739, 794, 849, 935, 994
HARSHAD AND MORAN NUMBERS
I've written about Harshad Numbers in the following posts:
In recreational mathematics, a Harshad number (or Niven number) in a given number base, is an integer that is divisible by the sum of its digits when written in that base. Harshad numbers in base n are also known as n-harshad (or n-Niven) numbers. Harshad numbers were defined by D. R. Kaprekar, a mathematician from India. The word "harshad" comes from the Sanskrit harṣa (joy) + da (give), meaning joy-giver. The term “Niven number” arose from a paper delivered by Ivan M. Niven at a conference on number theory in 1977.
They are quite frequent and account for about 12% of all the numbers up to 100,000. Here are the first few:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, 27, 30, 36, 40, 42, 45, 48, 50, 54, 60, 63, 70, 72, 80, 81, 84, 90, 100, 102, 108, 110, 111, 112, 114, 117, 120, 126, 132, 133, 135, 140, 144, 150, 152, 153, 156, 162, 171, 180, 190, 192, 195, 198, 200, 201, 204
If the dividend happens to be a prime number then the number is said to be a Moran number. To quote from Numbers Aplenty again:
A number \(n\) is a Moran number if \(n\) divided by the sum of its digits gives a prime number. For example, 111 is a Moran number because 111/(1+1+1) = 37 and 37 is a prime number. Moran numbers are a subset of Harshad numbers.
The first few Moran numbers are:
18, 21, 27, 42, 45, 63, 84, 111, 114, 117, 133, 152, 153, 156, 171, 190, 195, 198, 201, 207, 209, 222, 228.
MAGNANIMOUS NUMBERS
I've written about these in an eponymous post December 27th 2020. To quote from Numbers Aplenty, a magnanimous number can be define as:
A number (which we assume of at least 2 digits) such that the sum obtained inserting a "+" among its digit in any position gives a prime.
For example, 4001 is magnanimous because the numbers 4+001=5, 40+01=41 and 400+1=401 are all prime numbers.
Since all the prime numbers are odd, except for 2, all the magnanimous numbers, except for 11, are either a sequence of odd digits followed by an even digit, or a sequence of even digits followed by an odd digits.
It is conjectured that the magnanimous numbers are finite and that probably the largest one is 97393713331910, while the largest one which is also a prime number itself is probably 608844043.
The first such numbers are:
11, 12, 14, 16, 20, 21, 23, 25, 29, 30, 32, 34, 38, 41, 43, 47, 49, 50, 52, 56, 58, 61, 65, 67, 70, 74, 76, 83, 85, 89, 92, 94, 98, 101, 110, 112, 116, 118, 130, 136, 152, 158, 170, 172, 203
DIGITAL ROOT
While I've not made a specific post about digital roots, I've nonetheless mentioned them in the following posts:
The digital root (also repeated digital sum) of a natural number in a given radix is the (single digit) value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute a digit sum. The process continues until a single-digit number is reached. In base 10, this is equivalent to taking the remainder upon division by 9 (except when the digital root is 9, where the remainder upon division by 9 will be 0).
Associated with the digital root is the concept of additive persistence defined as:
The additive persistence counts how many times we must sum its digits to arrive at its digital root. For example, the additive persistence of 2718 in base 10 is 2: first we find that 2 + 7 + 1 + 8 = 18, then that 1 + 8 = 9.
SMITH AND HOAX NUMBERS
I mentioned Smith numbers in a blog post dating back to April 21st 2016 and titled Repunits and Smith Numbers.
Smith numbers are composite numbers with the property that the sum of their digits equals the sum of digits of their prime factors e.g. 22 → 2 + 2 = 4 and 22 = 2 * 11 → 2 + 1 + 1 = 4.
Hoax numbers are similar except they only consider distinct prime factors. Thus, for example, the Smith numbers 4 and 27 are excluded because the sums of their distinct prime factors are 2 and 3 respectively whereas their sums of digits are 4 and 9. The set of hoax numbers is a subset of the set of Smith numbers.
666 is a Smith number since 666 = 2 * 3 * 3 * 37 and 6 + 6 + 6 = 2 + 3 + 3 + 3 + 7.
The initial Smith numbers are:
4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 378, 382, 391, 438
The initial hoax numbers are:
22, 58, 84, 85, 94, 136, 160, 166, 202, 234, 250, 265, 274, 308, 319, 336, 346, 355
THE RATS SEQUENCE
I wrote about this in an eponymous post from September 26th 2020. Here is an excerpt:
A sequence produced by the instructions "reverse, add to the original, then sort the digits." For example, after 668, the next iteration is given by
668+866=1534
so the next term is 1345.
Applied to 1, the sequence gives:
1, 2, 4, 8, 16, 77, 145, 668, 1345, 6677, 13444, 55778, 133345, 666677, 1333444, 5567777, 12333445, 66666677, 133333444, 556667777, 1233334444, 5566667777, 12333334444, 55666667777, 123333334444, 556666667777, 1233333334444, ... (OEIS A004000).
Conway conjectured that an initial number leads to a divergent period-two pattern (such as the above in which the numbers of threes and sixes in the middles of alternate terms steadily increase) or to a cycle (Guy 2004, p. 404).
The lengths of the cycles obtained by starting with n= 1, 2, ... are 0, 0, 8, 0, 0, 8, 0, 0, 2, 0, ... (OEIS A114611), where a 0 indicates that the sequence diverges.
No comments:
Post a Comment