A Prime to Remember

Primes come and go but lately, as I keep a daily track of the number of my diurnal days, there has been more than usual. To illustrate, days 25447, 25453, 25457, 25463, 25469, and 25471 are all primes in a 6-4-6-6-2 pattern. After 25471 there will quite a drought because the next prime is 25523, a gap of 32.

Today I'm 25463 days old and I can't let it pass without recording some of its more interesting properties. One of these is that it is a member of OEIS A165572: the greater prime factor of successively better Golden Semiprimes. These semiprimes p*q, starting from 6=2*3, have the property that each successive value of q/p gives a better approximation of the Golden Ratio than the previous term where the $$ \text{Golden Ratio } \phi=\frac{1+\sqrt(5)}{2} \approx \, 1.61803398874989$$Here are the initial members of this sequence: 3, 5, 11, 31, 37, 47, 157, 571, 911, 1021, 1487, 2351, 3571, 24709, 25463. The corresponding semiprimes form OEIS A165570 and consist of 6, 15, 77, 589, 851, 1363, 15229, 201563, 512893, 644251, 1366553, 3416003, 7881197, 377331139, 400711231, 2963563859, 4035221017.

Here are the progressively better approximations as the larger factor of the semiprime is divided by the smaller:

3/2         1.50000000000000

        5/3         1.66666666666667

        11/7         1.57142857142857

    31/19         1.63157894736842

      37/23         1.60869565217391

     47/29         1.62068965517241

157/97         1.61855670103093

  571/353         1.61756373937677

    911/563         1.61811722912966

1021/631         1.61806656101426

1487/919         1.61806311207835

 2351/1453         1.61803165863730

  3571/2207         1.61803352967830

 24709/15271         1.61803418243730

25463/15737         1.61803393276991

Another property of 25463, albeit a base dependent one, is its membership in OEIS A156119: primes formed by rearranging five consecutive decimal digits (avoiding leading 0). No primes can be formed from {1,2,3,4,5} or {4,5,6,7,8} since they are divisible by three. Sequence is finite, ending with a(52)=96857. Initial members of sequence are: 10243, 12043, 20143, 20341, 20431, 23041, 24103, 25463.

Yet another property, again base dependent, is its membership of OEIS A124629: primes p such that their cubes are pandigital, meaning all digits from 0 to 9 must appear at least once; here 25463^3=16509301927847. The initial members of this sequence are: 5437, 6221, 7219, 8443, 10903, 11353, 15937, 17123, 18229, 19429, 20353, 20903, 20929, 21803, 21841, 21961, 22123, 22283, 22993, 23053, 23369, 23663, 24733, 25183, 25219, 25463.

Not base dependent is the property that 25463 shares as a member of OEIS A226154: smallest of four consecutive primes whose sum is a triangular number. Triangular numbers are of the form:$$ \binom{n}{2}= \frac{n \, (n-1)}{2}$$The initial members of this sequence are: 5, 23, 191, 389, 449, 2593, 3011, 5167, 5639, 5851, 8669, 18839, 25463. Here the four primes add to 101926 = 25463+25469+25471+25523 and this sum is a triangular number because: $$101926 = \binom{452}{2}=\frac{452 \times 451}{2}$$ 

Finally and again base independently, 25463 is a member of OEIS A022121: Fibonacci sequence beginning 3, 8. The initial members of this sequence are: 3, 8, 11, 19, 30, 49, 79, 128, 207, 335, 542, 877, 1419, 2296, 3715, 6011, 9726, 15737, 25463.

Random Walks

Let's consider the following situation. We start at the origin (0,0) and want to get to the point (4,4). However, we can only move one step at a time, either horizontally or vertically. We are constrained to move within the grid of points shown. Given this constraint, horizontal movement can be to the left or right and vertical movement can be up or down. However, we have no control over this step by step movement. It is completely random. On average, how many steps should be required to reach our destination?

FIGURE 1

I set up a program in SageMathCell to simulate this random walk over 1,000 trials. The result returned a median walk of 60 steps. What happens as the grid grows larger? I was interested in looking at the relationship between the size of the grid and the average number of steps required to reach the goal. Here are the results for grids with of size 1 to 21 and a graphical representation in FIGURE 2:

1	2	3	4	5	6	7	8	9	10	11
2	14	34	60	108	156	224	289	388	534	587

12	13	14	15	16	17	18	19	20	21
762	884	1104	1270	1420	1657	1785	2114	2442	2693

FIGURE 2

Not surprisingly the graph seems to be that of a parabola and my best fit formula, based on the above data, gives its equation as \(y=5.2 \, x^2\). 

This type of walk can be extended to 3 dimensions so that from (0, 0, 0) we need to get to (2, 2, 2) for example:

FIGURE 3

Running a thousand trials again on SageMathCell again, we get a median of 40 steps with a minimum of 6 (the least possible) and a maximum of 344. What's surprising is the vastly different lengths of these random walks. For example, with a cube of side 10, a median of 3075 steps is returned from the thousand trials but the maximum is 46826 and the minimum is 114. Here are the results (the simulation was too slow for sides greater than 10):

1	2	3	4	5	6	7	8	9	10
7	40	117	254	473	793	1129	1786	2390	3075

FIGURE 4

Although it looks parabolic, it's probably cubic and, if this is the case, then an equation of \(y=0.27 \, x^3 \) seems the best fit. In any case, this post is not meant to be definitive. It's just meant to clarify my thinking. I'll need to pursue this further and improve on the accuracy of these possible equations.

Primitive Abundant Numbers

Preliminary note: I've written about odd primitive abundant numbers in an earlier, eponymous post from May 21st 2017, so some content from that post is repeated here but there is new content as well. Here is the link.

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The sum of the proper divisors of an abundant number is greater than the number itself. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16. So what is a primitive abundant number?

To quote from Numbers Aplenty:

An abundant number is called primitive if none of its proper divisors is abundant. 

There are infinitely many such numbers, both even and odd. However Dickson proved that there are only a finite number of odd primitive abundant numbers with a given number of distinct prime factors. 

For example, there are only 8 odd primitive abundant numbers with 3 distinct prime factors, namely, 945, 1575, 2205, 7425, 78975, 131625, 342225, and 570375. 

The first primitive abundant numbers are 12, 18, 20, 30, 42, 56, 66, 70, 78, 88, 102, 104, 114, 138, 174, 186, 196 more terms. 

A second definition of primitive numbers excludes also those that have perfect proper divisors, like all multiples of 6. The first such numbers are 20, 70, 88, 104, 272, 304, 368, 464, 550, 572, 650, 748, 836, 945, 1184, 1312, 1376, 1430, 1504, 1575, 1696, 1870, 1888, 1952, 2002.

Here are some properties of primitive abundant numbers taken from Wikipedia:

Every multiple of a primitive abundant number is an abundant number. 

Every abundant number is a multiple of a primitive abundant number or a multiple of a perfect number. 

Every primitive abundant number is either a primitive semiperfect (also called primitive pseudoperfect) number or a weird number. 

There are an infinite number of primitive abundant numbers. 

The number of primitive abundant numbers less than or equal to \(n\) is \( o \left( \frac{n}{\log^2(n)} \right)\ \). 

A semiperfect or pseudoperfect number is a natural number that is equal to the sum of all or some of its proper divisors. A primitive semiperfect number (also called a primitive pseudoperfect number, irreducible semiperfect number or irreducible pseudoperfect number) is a semiperfect number that has no semiperfect proper divisor. The first few primitive semiperfect numbers are 6, 20, 28, 88, 104, 272, 304, 350, ... There are infinitely many odd primitive semiperfect numbers, the smallest being 945.

A weird number is a natural number that is abundant but not semiperfect or pseudoperfect. In other words, the sum of the proper divisors (divisors including 1 but not itself) of the number is greater than the number, but no subset of those divisors sums to the number itself. The first few weird numbers are 70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, ...

See my blog post titled Zumkellar, Half-Zumkellar, and Pseudoperfect Numbers and Odd Primitive Abundant Numbers.

Admirable Numbers and Compatible Numbers

Yesterday I turned 25446 days and this number was identified by Numbers Aplenty as an admirable number, defined as a number \(n\) for which there exists a divisor \(d\) of \(n\) such that \(2n = \sigma(n)-2d\). In other words, \(n\) is equal to the sum of its proper divisors, where one of them has a minus sign.

For 25446, the divisors are: 1, 2, 3, 6, 4241, 8482, 12723, 25446 and the sum of these divisors is 50904. However, 50904 - 2 x 6 = 50892 = 2 x 25446 and here the divisor 6 has been assigned the minus sign. The modified divisors (1, 2, 3, -6, 4241, 8482, 12723) now add to 25446. The previously mentioned website goes on to say that:

Clearly, admirable numbers are a subset of abundant numbers and they are infinite because, for example, all the numbers 6\(p\), with \(p\)>3 prime, are admirable. The largest number that cannot be written as a sum of admirable numbers is 1003. Pairs of consecutive admirable numbers are rarer than pairs of consecutive abundant numbers. Up to \(10^{12}\), there are only two such pairs, namely 29691198404, 29691198405 and 478012798575, 478012798576.

On the other hand, pairs of admirable numbers that differ by two are more common but still sparse. There are 72 such pairs up to 27000.

 

The smallest 3 x 3 magic square made up of admirable numbers is shown in Figure 1.

Figure 1: smallest possible magic square
made from admirable numbers

OEIS A111592 lists the initial admirable numbers:

12, 20, 24, 30, 40, 42, 54, 56, 66, 70, 78, 84, 88, 102, 104, 114, 120, 138, 140, 174, 186, 222, 224, 234, 246, 258, 270, 282, 308, 318, 354, 364, 366, 368, 402, 426, 438, 464, 474, 476, 498, 532, 534, 582, 606, 618, 642, 644, 650, 654, 672, 678, 762, 786, 812, ...

These numbers are related to Zumkellar, Half-Zumkellar and pseudoperfect numbers in that they all involve the divisors of the number. See my blog post on these sorts of numbers.

In the OEIS comments, we read that "the concept of admirable numbers was developed by educator Jerome Michael Sachs (1914-2012) for a television in-service training course in mathematics for elementary school teachers." Here is the link to the article that he wrote in The Arithmetic Teacher, Vol. 7, No. 6 (1960), pp. 293-295. However, in that article he allows more than one of the divisors of a number to be negative. For example, he writes 24 as being equal to the following algebraic sum of its divisors: 4+6+8+12-1-2-3. However, this is the same as 1+2+3+4+8+12-6 so it's not clear whether a sum involving multiple negative divisors is always equivalent to another sum involving a single negative divisor.

Sachs also introduces the notion of a compatible number pair as an extension or relaxation of the concept of an amicable number pair. For example, 220 and 284 are an amicable pair because the proper divisors of each add to the other number. The proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110 and these add to 284. The proper divisors of 284 are 1, 2, 4, 71 and 142 and these add to 220. In such cases, the smaller number is abundant and the larger number deficient.

Sach's proposal for a compatible number pair is two numbers such that the algebraic sums of their divisors each leads to the other number. For example:

• 30 has divisors of 1, 2, 3, 5, 6, 10, 15
• 40 has divisors of 1, 2, 4, 5, 8, 10 and 20
• 40 = 2 + 3 + 5 + 6 + 10 + 15 - 1
• 30 = 1 + 2 + 4 + 5 + 8 + 20 - 10

So he defines 30 and 40 as compatible numbers.

The smaller members of such pairs are listed in OEIS A109797 while the larger members are listed in OEIS A109798.

Here is the SageMath code to generate the admirable numbers between 25000 and 26000

The Apocryphal Diderot-Euler Encounter

There is an interesting story about an encounter between Diderot and Euler in the palace of Catherine the Great in St.Peterburg. I've come across two versions of the story, one in Bell's "Men of Mathematics" and the other in Hogben's "Mathematics for the Million". Both are essentially the same and there are many other slightly differing versions about. Here is the account by Hogben:

Figure 1

This is nonsense because Diderot was an accomplished mathematician in his own right. He apparently didn't know how to respond and, embarrassed, made a quick exit. The next day he asked the Empress for safe passage back to Paris. Even though the story is apocryphal, the mathematical equation interested me (from a mathematical perspective not a theological one), so I thought I'd investigate it a little. Firstly though I imposed some restriction on a, b and n: they must be integers and all greater than zero. $$  \frac{a+b^n}{n}=x \text{   with }a, b, c >0 \text { and } a, b, c \, \in \, \, Z $$Let's consider the case where \(x=100\) and \( n=1\). We have simply:$$ a+b=100 \text{ and }b=100-a$$Thus \(a=1\) and \(b=99 \), \(a=2 \) and \(b=98 \), ..., \(a=99\) and \(b=1\) are the possible solutions.

Let's next consider the case where \(x=100\) but \(n=2\). In this case we get:$$ \frac{a+b^2}{2}=100 \text{ and }b=\sqrt{200-a}$$For this result, the values of a must be chosen so that \(\sqrt{200-a} \) is a square number. The square numbers between 0 and 200 are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196 and correspond to \(b\) values of 1, 2, ..., 13, 14 with associated \(a\) values of 199, 196, ..., 31, 4.

In the case where \(x=100\) and \(n=3\) we get: $$ \frac{a+b^3}{3}=100 \text{ and }b=\sqrt[3]{300-a}$$Here, the values of a must be chosen so that we can find an integral cube root of the number under the cube root sign. The cubes that lie between 0 and 300 are 1, 8, 27, 64, 125 and 216 and so \(a\) values of 299, 292, 273, 236, 175 and 84 correspond to \(b\) values of 1, 2, 3, 4, 5 and 6.

As n increases, the possible values of \(a \) decrease:

• fourth power (\(n=4\)) numbers less than 400 are 1, 16, 81 and 256
• fifth power numbers (\(n=5\)) less than 500 are 1, 32 and 243
• sixth power numbers (\(n=6\)) less than 600 are 1 and 64
• seventh power numbers (\(n=7\)) less than 700 are 1 and 128
• eighth power numbers (\(n=8\)) less than 800 are 1 and 256
• ninth power numbers (\(n=9\)) less than 900 are 1 and 512
• tenth power numbers (\(n=10\)) less than 1000 are 1 only

As can be seen, the multiples of 100 are soon overtaken. So just to check, let's take the case where n=9 and we want \(900-a=512\) and so \(a=388\) and \(b=2\). This means 100 can be written as:$$100=\frac{388+2^9}{9}$$From this analysis, it's clear that for any given integer \(x\) there are numerous ways to represent it in the form examined but their number is definitely finite. 

Hogben Numbers

Today I turned 25441 days and one of the properties of this number is that it's a Hogben number, the 160th such number in fact. I discovered this fact thanks to Numbers Aplenty in that site's description of the properties of this number. A link provided there contains the following information about these sorts of numbers:

The \(n\)-th Hogben number \(H_n\) is equal to \(n^2-n+1\).

Figure 1: SPIRAL ARRANGEMENT OF INTEGERS

In a spiral arrangement of the integers, Hogben numbers appear on the main diagonal (see Figure 1). Hogben numbers are often called central polygonal numbers. \(H_n\) is also the maximal number of ones that a \(n \times n\) {0,1} matrix can contain and still be invertible (that is an inverse matrix exists). The first Hogben numbers are 1, 3, 7, 13, 21, 31, 43, 57, 73, 91, 111, 133, 157, 183, 211, 241, 273, 307, 343, 381, ...

The Hogben numbers are listed in OEIS A002061 as the central polygonal numbers: \(a(n) = n^2 - n + 1\) with some interesting comments about where these Hogben numbers crop up. Let's consider some examples:

EXAMPLE 1

For n>1: a(n) is the maximum total number of queens that can coexist without attacking each other on an [n+1] x [n+1] chessboard. Specifically, this will be a lone queen of one colour placed in any position on the perimeter of the board, facing an opponent's "army" of size a(n)-1. 

The normal chess board consists of 8 x 8 = 64 squares, so the value of \(n^2-n+1\) when \(n=7\), namely 43, will give the maximum possible number of such queens. Figure 2 confirms this to be the case:

Figure 2: one against 42

EXAMPLE 2

Since \( (n+1)^2 - (n+1) + 1 = n^2 + n + 1 \) then from 7 onwards these are also exactly the numbers that are represented as 111 in all number bases: \(111_2=7 \), \(111_3=13 \), ... 

This is quite an interesting property and so we have the result that:

111 in base 2 is 7
111 in base 3 is 13
111 in base 4 is 21
111 in base 5 is 31
111 in base 6 is 43
111 in base 7 is 57
111 in base 8 is 73
111 in base 9 is 91
111 in base 10 is 111
111 in base 11 is 133
111 in base 12 is 157
111 in base 13 is 183
111 in base 14 is 211
111 in base 15 is 241
111 in base 16 is 273

The next example is taken from this source:

EXAMPLE 3

Figure 3: Hogben numbers within an isosceles triangle of numbers

This is a nice visual result and reveals an easy way to generate these numbers in a manner similar to Pascal's triangle.

So what about Lancelot Thomas Hogben after whom these numbers were named. He turns out to have been quite an interesting character. To quote from his Wikipedia entry:

Lancelot Thomas Hogben (9 December 1895 – 22 August 1975) was a British experimental zoologist and medical statistician. He developed the African clawed frog (Xenopus laevis) as a model organism for biological research in his early career, attacked the eugenics movement in the middle of his career, and popularised books on science, mathematics and language in his later career.

It is in his very popular 1936 book about Mathematics titled "Mathematics for the Million" that he presumably deals with numbers of the form \(n^2-n+1\). I have a copy of this book in electronic format in my library of ebooks but I couldn't find anything that sheds further light on these particular numbers. Here is a quote from the introduction to his book:

Figure 4: Lancelot Thomas Hogben 

Hogben was a conscientious objector in World War 1 and was imprisoned for a time. He was fiercely opposed to the Eugenics movement that was very active in the 1920s and 1930s. In World War 2, he was responsible for the British Army's medical statistics. He is quoted as saying:

"I like Scandinavians, skiing, swimming and socialists who realise it is our business to promote social progress by peaceful methods. I dislike football, economists, eugenicists, Fascists, Stalinists, and Scottish conservatives. I think that sex is necessary and bankers are not".

In addition to his extensive writing, he edited The "Loom of Language" by his friend Frederick Bodmer. This was a book that I borrowed from the library in Pontefract, Yorkshire, when I lived there in 1983-4, and which I read and very much enjoyed.

Augustus De Morgan

Augustus De Morgan was a British mathematician and logician. He formulated De Morgan's laws and introduced the term mathematical induction, making its idea rigorous. He famously stated that he was \(x\) years old in the year \(x^2\), leaving us to surmise when he was born. A definite answer is possible once we know that he was born in the 19th century. This is a laughably simple problem and I am not suggesting that it has any mathematical significance but it popped up as an exercise in a book that I just started reading titled "Elementary Number Theory with Applications" by Thomas Koshy. It inclined me to find out a little more about this mathematician but first let's deal with the problem, simple as it might be.

One approach is to find a number between 1801 and 1900 that is a square number. There is only one such number and that is \(1849=43^2\). Thus we can say that he was born in 1806 and indeed he was born on the 27th of June 1806 and died on the 18th March 1871). This leads us to ask what in the next birth year that would allow its natives to make a similar claim. Well, those who were 44 years old in 1936 could make such a claim since \(1936=44^2\) and all would have been born in the 1892. Similarly, anyone born in 1980 will turn 45 in the year \(2025=45^2\).

The same approach could be taken with the cube of the year. If one were 12 years old in 1728, the claim could be made that one was \(x\) years old in the year \(x^3\). One would have be 13 years old in 2197 to make the same claim. The years that are perfect cubes will obviously be much farther apart than the perfect squares.

What about De Morgan himself? The Wikipedia article seems to give the most comprehensive account of his life. I was reminded that I had a copy of E. T. Bell's "Men of Mathematics" but De Morgan doesn't get a mention in that. He was a confirmed athiest:

His mother was an active and ardent member of the Church of England, and desired that her son should become a clergyman, but by this time De Morgan had begun to show his non-conforming disposition. 

As he himself said in 1838:

There is a word in our language with which I shall not confuse this subject, both on account of the dishonourable use which is frequently made of it, as an imputation thrown by one sect upon another, and of the variety of significations attached to it. I shall use the word Anti-Deism to signify the opinion that there does not exist a Creator who made and sustains the Universe. 

Even though he obtained a Bachelor of Arts degree at Cambridge, he could not progress to a Master's degree because that involved a theological test to which De Morgan would not subject himself to (even though he had been brought up in the Church of England).

As no career was open to him at his own university, he decided to go to the Bar, and took up residence in London; but he much preferred teaching mathematics to reading law. About this time the movement for founding London University (now University College London) took shape. The two ancient universities of Oxford and Cambridge were so guarded by theological tests that no Jew or Dissenter outside the Church of England could enter as a student, still less be appointed to any office. A body of liberal-minded men resolved to meet the difficulty by establishing in London a University on the principle of religious neutrality. De Morgan, then 22 years of age, was appointed professor of mathematics.

The theological test for Oxford and Cambridge was abolished in 1875. De Morgan was an outstanding teacher of Mathematics as well as a brilliant and witty writer. He was a lifelong friend of the Irish mathematician William Rowan Hamilton who discovered the Quaternions.

Of his childhood:

Augustus De Morgan was born in Madurai, India in 1806.[a] His father was Lieut.-Colonel John De Morgan (1772–1816), who held various appointments in the service of the East India Company. His mother, Elizabeth Dodson (1776–1856), was a descendant of James Dodson, who computed a table of anti-logarithms, that is, the numbers corresponding to exact logarithms. Augustus De Morgan became blind in one eye a month or two after he was born. The family moved to England when Augustus was seven months old. As his father and grandfather had both been born in India, De Morgan used to say that he was neither English, nor Scottish, nor Irish, but a Briton "unattached", using the technical term applied to an undergraduate of Oxford or Cambridge who is not a member of any one of the Colleges.

In Autumn of 1837, he married Sophia Elizabeth Frend (1809–1892). Of his family:

De Morgan had three sons and four daughters, including fairytale author Mary de Morgan. His eldest son was the potter William De Morgan. His second son George acquired distinction in mathematics at University College and the University of London. He and another like-minded alumnus conceived the idea of founding a mathematical society in London, where mathematical papers would be not only received (as by the Royal Society) but actually read and discussed. The first meeting was held in University College; De Morgan was the first president, his son the first secretary. It was the beginning of the London Mathematical Society. 

Unfortunately, his son George (the previously mentioned first secretary of the London Mathematical Society) died and not long after a daughter. After this, his health deteriorated and he died of "nervous prostration" at age 64.

De Morgan also promoted the work of the self-taught Indian mathematician Ramchundra. Here is an excerpt from the Wikipedia article about Ramchundra:

Ramchundra (1821–1880) was a British Indian mathematician. His book, Treatise on Problems of Maxima and Minima, was promoted by the prominent mathematician Augustus De Morgan. In his introduction to Ramchundra's book, De Morgan says that he was born in 1821 in Panipat to Sunder Lal, a Kayasth of Delhi. De Morgan came to know of Ramchundra when, in 1850, he was sent by a friend to work on maxima and minima by the 29-year-old self-taught mathematician. Ramchundra had published his book at his own expense in Calcutta in that year. De Morgan arranged for the book to be republished in London under his own supervision. De Morgan was so impressed that he undertook to bring Ramchundra's work to the notice of scientific men of Europe. Charles Muses, in an article in the Mathematical Intelligencer (1998) called Ramchundra "De Morgan's Ramanujan". He was mystified why, in spite of De Morgan's efforts to make this "remarkable Hindu algebraist known, he does not appear in most texts on history of mathematics." Ramchundra was teacher of science in Delhi College for some time. In 1858, he was native head master in Thomason Civil Engineering College (now Indian Institute of Technology, Roorkee) at Roorkee. Later that year, he was appointed head master of a school in Delhi.

Zumkellar Numbers, Half Zumkellar Numbers and Pseudoperfect Numbers

Zumkeller numbers are related to perfect numbers such as 6, the divisors of which can be written as the set {1, 2, 3, 6}. There are two mutually exclusive subsets of this set, {1, 2, 3}  and {6}, whose union is the original set and both of whose elements add to 6. Zumkeller numbers are similar in that there are two mutually exclusive subsets of this set of divisors, whose union is the original set but with the difference that both of the elements in each subset add to a number other than the originating number. For example, 20 has the set of divisors {1, 2, 4, 5, 10, 20} and there are two subsets {1, 20} and {2, 4, 5, 10} that both total 21. Thus 20 is a Zumkellar number. Of course \( \sigma(20) \), the sum of the divisors of 20, is 42 and so each subset sums to half of that.

If \( n \) is a Zumkeller number, then \( \sigma(n) \) is even and \(n\) is perfect or abundant. A number is abundant if the sum of its proper divisors is greater than the number. All the practical numbers \( n \), with \( \sigma(n) \) even, are also Zumkeller numbers. Here is a link to my blog post on practical numbers.

Bhakara Rao & Peng have proved several results on Zumkeller numbers such as the fact that \(n!\) is a Zumkeller numbers for \(n\ge 3 \). For example, 5! =120 and the set of divisors is {1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120}. This set can be divided into {60, 120} and {1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40}, both of which total 180. For what it's worth, the following YouTube video features Bhakara Rao introducing the paper that he wrote with Peng.

OEIS A083207 lists the initial Zumkeller numbers:

6, 12, 20, 24, 28, 30, 40, 42, 48, 54, 56, 60, 66, 70, 78, 80, 84, 88, 90, 96, 102, 104, 108, 112, 114, 120, 126, 132, 138, 140, 150, 156, 160, 168, 174, 176, 180, 186, 192, 198, 204, 208, 210, 216, 220, 222, 224, 228, 234, 240, 246, 252, 258, 260, 264, 270, 272, ...

In the OEIS comments for this sequence, it's stated that:

The 229026 Zumkeller numbers less than one million have a maximum difference of 12. This leads to the conjecture that any 12 consecutive numbers include at least one Zumkeller number. There are 1989 odd Zumkeller numbers less than one million; they are exactly the odd abundant numbers that have even abundance. 

Today, my diurnal age is 25434, which is a Zumkellar number, and the next one is 25440. This is a difference of only six. From the earlier statistics, it can be seen that Zumkeller numbers occur with a frequency of about 22.9% for the first one million natural numbers.

Now OEIS A246198 defines half-Zumkeller numbers as numbers whose proper positive divisors can be partitioned into two disjoint sets whose sums are equal. The comment is made that all even half-Zumkeller numbers are in OEIS A083207, i.e. they are Zumkeller numbers. The first 47 terms coincide with A083207. 225 is the first number in the sequence that is not a Zumkeller number. The set of proper divisors of 225 is {1, 3, 5, 9, 15, 25, 45, 75} and this can be divided into two disjoint sets, {9, 75, 5} and {25, 3, 1, 45, 15} both totalling 89.

OEIS A005835 describes the pseudoperfect (or semiperfect) numbers as those in which some subset of the proper divisors of n sums to n. It's noted in the comments that deficient numbers cannot be pseudoperfect and that the first odd pseudoperfect number is 945. In the case of 945, the proper divisors are {1, 3, 5, 7, 9, 15, 21, 27, 35, 45, 63, 105, 135, 189, 315} and there are five subsets of these divisors that add to 945. Here is the list:

• {1, 9, 21, 27, 35, 45, 63, 105, 135, 189, 315}
• {3, 7, 21, 27, 35, 45, 63, 105, 135, 189, 315}
• {7, 9, 15, 27, 35, 45, 63, 105, 135, 189, 315}
• {1, 3, 5, 7, 15, 27, 35, 45, 63, 105, 135, 189, 315} 
• {1, 5, 7, 9, 15, 21, 35, 45, 63, 105, 135, 189, 315} 

While nearly all abundant numbers are pseudoperfect, some aren't. These numbers are termed weird and comprise OEIS A006037: weird numbers - abundant (A005101) but not pseudoperfect (A005835). From the comments to this sequence in the OEIS, we find:

Deléglise (1998) shows that abundant numbers have asymptotic density < 0.2480, resolving the question which he attributes to Henri Cohen of whether the abundant numbers have density greater or less than 1/4. The density of pseudoperfect numbers is the difference between the densities of abundant numbers (A005101) and weird numbers (A006037), since the remaining integers are perfect numbers (A000396), which have density 0. Using the first 22 primitive pseudoperfect numbers (A006036) and the fact that every multiple of a pseudoperfect number is pseudoperfect it can be shown that the density of pseudoperfect numbers is > 0.23790.

There are other interesting facts mentioned in the comments, including:

• The first weird number that has more than one decomposition of its divisors set into two subsets with equal sum (and thus is not a member of A083209) is 10430:

  1+5+7+10+14+35+298+10430 = 2+70+149+745+1043+1490+2086+5215

  2+70+298+10430 = 1+5+7+10+14+35+149+745+1043+1490+2086+5215.

• A weird number n multiplied with a prime \( p > \sigma(n) \) is again weird. Primitive weird numbers (A002975) are those which are not a multiple of a smaller term, i.e., don't have a weird proper divisor.
• No odd weird number exists below \(10^{21}\).

One thing to bear in mind about all the types of numbers mentioned above is that they are independent of the number system base used to depict the numbers. Other types of numbers are dependent on the number base. For example, a d-powerful number in base 10 is not necessarily a d-powerful number in another base.

Neil Sloane, the originator of the OEIS, has this to say about Reinhard Zumkellar, after whom the numbers are named:

I am deeply sorry to have to report that Reinhard Zumkeller passed away at the end of March 2016. He suffered from pancreatic cancer, which had already progressed to an advanced stage when it was diagnosed. He was a long-time contributor to the OEIS, and was later an editor and then a diligent and dedicated editor-in-chief. Between 2000 and 2016 he contributed over 23000 items to the OEIS. Reinhard was a great Haskell expert, and he was already ready to write a Haskell program and compute 10000 terms when I was studying a new sequence and wanted to see a graph. He will be greatly missed. Neil Sloane, July 3, 2016. 

Platonic Numbers

Today I turned 25432 days old and I've mentioned in my previous post that this number is d-powerful in five different ways and part of a string of ten consecutive d-powerful numbers, starting with 25430 and ending with 25439. However, I also discovered that 25432 is a Platonic number defined as follows:

Platonic numbers (A053012) are the numbers of dots in a layered geometric arrangement into one of the 5 Platonic solids. The platonic numbers start with one initial dot (for n=1), then with one dot at each vertex of a given Platonic solid (for n=2), with each of the following layers growing out of the initial vertex with one more dot per edge than the preceding layer, and where overlapping dots (the dot at the initial vertex and the dots on all the edges sharing that initial vertex) are counted only once.

The 5 types of Platonic numbers (by increasing number of vertices) are:

A000292: Tetrahedral (or triangular pyramidal) numbers: $$ \binom{n + 2}{3} = \frac{n \, (n + 1) \, (n+2)}{6} $$ A005900: Octahedral numbers: $$ \frac{2n^3+n}{3}=\frac{n \, (2n^2+1)}{3} $$ A000578: The Cubes: $$n^3 $$A006564: Icosahedral numbers: $$ \frac{n \, (5n^2-5n+2)}{2} $$A006566: Dodecahedral numbers: $$ \frac{n \,(9n^2-9n + 2)}{2}=\frac{n \, (3n- 1) \, (3n-2)}{2}=\frac{3n \, (3n-1) \, (3n-2)}{6}=\binom{3n}{3} $$This YouTube video shows an animation of how the platonic numbers emerge from the five different solids:

In the case of 25432, my diurnal age on the 19th November 2018, it is an icosahedral number with the value of \( n \) being 22. It is the 133rd Platonic number with the next being 26214.

Additionally, there are the centered Platonic numbers defined by starting with 1 central dot (for n=0) and adding regular convex polyhedral layers around the central dot, where the nth layer, n ≥ 1, has n+1 dots per facet ridge (face edge for polyhedrons) including both end vertices. The formulae are very similar to the above and can be explored further here.

D-POWERFUL NUMBERS

The D-powerful numbers are those that can be expressed as the sum of their individual digits raised to positive powers. For example, today I am 25430 days old and this can be written as \( 2^{13} + 5^3 + 4^7 + 3^6 + 0 \) where the 0 of course can be raised to any positive power. Whenever a d-powerful number ends in a 0, it follows that all the subsequent numbers in that decade will also be d-powerful. Thus:

\( 25431 = 2^{13} + 5^3 + 4^7 + 3^6 + 1 \)

\(25432 = 2^{13} + 5^3 + 4^7 + 3^6 + 2 \)

\(25433 = 2^{13} + 5^3 + 4^7 + 3^6 + 3 \)

\(25434 = 2^{13} + 5^3 + 4^7 + 3^6 + 4 \)

\(25435 = 2^{13} + 5^3 + 4^7 + 3^6 + 5 \)

\(25436 = 2^{13} + 5^3 + 4^7 + 3^6 + 6 \)

\(25437 = 2^{13} + 5^3 + 4^7 + 3^6 + 7 \)

\(25438 = 2^{13} + 5^3 + 4^7 + 3^6 + 8 \)

\(25439 = 2^{13} + 5^3 + 4^7 + 3^6 + 9 \)

The occurrences of such decades are relatively frequent. There are 600 d-powerful numbers in the range from 1 to 17463 inclusive. This represents a relative frequency of a little over 3.4%. D-powerful numbers can be written in more than one way. For example, 25432 can be written in five different ways:

\(25432 = 2^{13} + 5^3 + 4^7 + 3^6 + 2 \)

\(25432 = 2^{12} + 5^4 + 4^5 + 3^9 + 2^2 \)

\(25432 = 2^{10} + 5^4 + 4^6 + 3^9 + 2^2 \)

\(25432 = 2^{11} + 5^5 + 4^3 + 3^9 + 2^9 \)

\(25432 = 2^{12} + 5^4 + 4 + 3^9 + 2^{10} \)

Similarly, 25437 can be written in two ways:

\(25437 = 2^{13} + 5^3 + 4^7 + 3^6+ 7\)

\(25437 = 2^{11} + 5 + 4^2 +3^8 + 7^5\)

Finally, 25439 can be written in two ways:

\(25439 = 2^{13} + 5^3 + 4^7 + 3^6 +9 \)

\(25439 = 2^{13} + 5^3 + 4^7 +3^2 + 3^3 \)

OEIS A007532 refers to d-powerful numbers as simply powerful numbers and lists the following initial member of the sequence:

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, 1034

So I have another nine d-powerful days to go. This website gives some examples of interesting d-powerful numbers with additional special properties. For example, consider the case of d-powerful palindromic primes. There is only one such 3 digit number, 373, and it has two representations:

\(373 = 3^1 + 7^3 + 3^3 \)

\(373 = 3^4 + 7^2 + 3^5 \)

There is only one 5 digit d-powerful palindromic prime and that is 98389:

\(98389 = 9^4 + 8^1 + 3^1 + 8^5 + 9^5 \)

Another category are d-powerful numbers that a palindromic numbers and have a representation where the exponents are symmetric. The first such numbers are:

\(262 = 2^7 + 6^1 + 2^7 \)

\(4224 = 4^3 + 2^{11} + 2^{11} + 4^3 \)

\(39393 = 3^9 + 9^1 + 3^2 + 9^1 + 3^9 \)

\(79597 = 7^1 + 9^3 + 5^7 + 9^3 + 7^1 \)

The smallest d-powerful, pandigital number is 1023456879 with representation:

\(1023456879 = 1^1 + 0^1 + 2^{27} + 3^{18} + 4^6 + 5^1 + 6^{11} + 8^9 + 7^5 + 9^7\)

Here is the SageMath code to determine the indices of a number if it is d-powerful (25430 is used as the test number but any number can be substituted for it, provided it has five digits:

Smith Brothers

2542825429

On Thursday, the 21st April 2106, I posted about Repunits and Smith Numbers. The day was 24490 and the number turned out to be a member of OEIS A104167, a sequence whose members have the property that, when multiplied by any repunit prime, the result is a Smith number. Just to recapitulate from that post:

A repunit is defined by Wikipedia as a number like 11, 111, or 1111 that contains only the digit 1 — a more specific type of repdigit. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler in his book "Recreations in the Theory of Numbers". A repunit prime is a repunit that is also a prime number. 

A Smith number is defined by Wikipedia as a composite number for which, in a given base (in base 10 by default), the sum of its digits is equal to the sum of the digits in its prime factorisation. For example, 378 = 2 × 3 × 3 × 3 × 7 is a Smith number since 3 + 7 + 8 = 2 + 3 + 3 + 3 + 7. In this definition the factors are treated as digits: for example, 22 factors to 2 × 11 and yields three digits: 2, 1, 1. Therefore 22 is a Smith number because 2 + 2 = 2 + 1 + 1. 

Today, I turned 25428 days old and this number is a Smith number with the property that the next consecutive number (25429) is also a Smith number. Such pairs of numbers are termed Smith brothers. They are not that common. OEIS A050219 lists the smaller of the members of each pair. Here is the list as shown on the OEIS website:

728, 2964, 3864, 4959, 5935, 6187, 9386, 9633, 11695, 13764, 16536, 16591, 20784, 25428, 28808, 29623, 32696, 33632, 35805, 39585, 43736, 44733, 49027, 55344, 56336, 57663, 58305, 62634, 65912, 65974, 66650, 67067, 67728, 69279, 69835

Here is a SageMathCell with the code that I wrote to generate this sequence (up to 25428). Note that print F must be changed to print(F) because SageMath is now using Python 3 and the old Python 2 code for print no longer works:

It works but I'm sure there are more elegant ways to generate the same result. Here is a Numberphile YouTube video that explains a little more about Smith Numbers:

2019: A Numerical Profile

With the coming year, 2019, less than two months away, I decided to investigate some of the numerical properties adhering to this number. Right off the bat, we can see that its digit sum is 12 and, because 3 divides 12, we know that 3 will divide 2019 as well. In fact, 2019 has prime factors of 3 and 673. Thus

2019 = 3 * 673

This means that 673 AD and 1346 AD (673 + 673 = 1346) could be associated with 2019 AD. 673 AD was a time of great expansion for the Islamic world (Muhammad had died in 632 AD). The following year, the Siege of Constantinople (one of many over the centuries) began, but "in 672–673 Arab fleets secured bases along the coasts of Asia Minor, and then proceeded to install a loose blockade around Constantinople." Here are some more details :

The First Arab Siege of Constantinople in 674–678 was a major conflict of the Arab–Byzantine wars, and the first culmination of the Umayyad Caliphate's expansionist strategy towards the Byzantine Empire, led by Caliph Mu'awiya I. Mu'awiya, who had emerged in 661 as the ruler of the Muslim Arab empire following a civil war, renewed aggressive warfare against Byzantium after a lapse of some years and hoped to deliver a lethal blow by capturing the Byzantine capital, Constantinople.

As reported by the Byzantine chronicler Theophanes the Confessor, the Arab attack was methodical: in 672–673 Arab fleets secured bases along the coasts of Asia Minor, and then proceeded to install a loose blockade around Constantinople. They used the peninsula of Cyzicus near the city as a base to spend the winter, and returned every spring to launch attacks against the city's fortifications. Finally, the Byzantines, under Emperor Constantine IV, managed to destroy the Arab navy using a new invention, the liquid incendiary substance known as Greek fire. The Byzantines also defeated the Arab land army in Asia Minor, forcing them to lift the siege. The Byzantine victory was of major importance for the survival of the Byzantine state, as the Arab threat receded for a time. A peace treaty was signed soon after, and following the outbreak of another Muslim civil war, the Byzantines even experienced a period of ascendancy over the Caliphate.

1346 AD was also an interesting year. As reported in Wikipedia, it included these events:

• in Spring, a severe Black Death epidemic began its spread at the River Don near the Black Sea, then spread throughout Russia, the Caucasus, and the Genovese provinces within the year
• on April 16th,  the Serbian Empire was proclaimed in Skopje by Dusan Silni, occupying much of South-Eastern Europe
• on July 11th and 12th, Edward III and the English army cross the English Channel, and begin an invasion of France
• on August 26, at the Battle of Crécy, the English defeat the French, in the first European battle where gunpowder is used.

Well, there's not much Mathematics in the history above so best to move on to more mathematical matters. What follows are some interesting facts about 2019 as number.

OEIS A037015: Numbers n with property that, reading binary expansion of n from right to left, run lengths strictly increase. Here we have \( 2019_{10}=11111100011_2 \). The initial members of the sequence are:

0, 1, 3, 6, 7, 14, 15, 28, 30, 31, 57, 60, 62, 63, 120, 121, 124, 126, 127, 241, 248, 249, 252, 254, 255, 483, 496, 497, 504, 505, 508, 510, 511, 966, 993, 995, 1008, 1009, 1016, 1017, 1020, 1022, 1023, 1987, 1990, 2016, 2017, 2019, 2032, 2033, 2040, 2041, 2044

OEIS A158339: Semiprimes that are the sum of four successive semiprimes. Here we have: 501 + 502 + 505 + 511 = 2019 and 501 = 3 * 167, 502 = 2 * 251, 505 = 5 * 101 and 511 = 7 * 73.

The initial members of this sequence are:

39, 94, 106, 118, 146, 158, 185, 201, 221, 254, 302, 365, 427, 473, 485, 519, 537, 589, 633, 655, 707, 723, 749, 767, 842, 851, 869, 901, 1003, 1145, 1205, 1211, 1219, 1247, 1263, 1337, 1349, 1603, 1646, 1681, 1703, 1731, 1797, 1891, 1903, 1937, 2005, 2019

OEIS A193227: Semiprimes p*q such that p+1 and q+1 are semiprimes. 

Here p+1 = 4 = 2 * 2 and q+1 = 674 = 2 * 337 and the initial members of this sequence are:

9, 15, 25, 39, 65, 111, 169, 183, 185, 219, 305, 365, 471, 481, 579, 785, 793, 831, 939, 949, 965, 1191, 1263, 1369, 1371, 1385, 1565, 1623, 1839, 1983, 1985, 2019

OEIS A091431: Happy-go-Lucky numbers: numbers that are both Happy (OEIS A007770) and Lucky (OEIS A000959). Happy numbers are those whose repeated sums  of squares of digits return 1.

The initial members of this sequence are:

1, 7, 13, 31, 49, 79, 129, 133, 193, 219, 319, 331, 367, 391, 409, 487, 655, 673, 739, 931, 937, 1009, 1029, 1039, 1093, 1209, 1233, 1251, 1275, 1281, 1285, 1303, 1309, 1323, 1339, 1533, 1575, 1587, 1599, 1663, 1771, 1857, 1933, 1959, 1995, 2019

OEIS A076408: Sum of first n perfect powers. As Wikipedia defines it: 

In mathematics, a perfect power is a positive integer that can be expressed as an integer power of another positive integer. More formally, n is a perfect power if there exist natural numbers m > 1, and k > 1 such that \( m^k = n \). In this case, n may be called a perfect k-th power. If k = 2 or k = 3, then n is called a perfect square or perfect cube, respectively. Sometimes 1 is also considered a perfect power (\(1^k = 1\) for any k).

Here n=22 and, as shown in OEIS A001597, the first 22 perfect powers are: 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, 216, 225, 243 with the sequence of progressive sums being: 1, 5, 13, 22, 38, 63, 90, 122, 158, 207, 271, 352, 452, 573, 698, 826, 970, 1139, 1335, 1551, 1776, 2019.

In fact, there are 140 sequences listed in the OEIS that contain a reference to 2019. I've covered some of the more interesting ones, or at least ones that I could understand.

From Numbers Aplenty, we have the following properties:

2019 is the smallest number that can be written in six ways as the sum of the squares of three primes. Here are the prime triplets:

7 11 43, 7 17 41, 11 23 37, 13 13 41, 17 19 37, 23 23 31

Figure 1 is an extract from the Numbers Aplenty page:

FIGURE 1