Aliquot Sequences

My attention was drawn to aliquot sequences today, day 25098, because the number is a member of four aliquot sequences (as shown below):

From any starting point, it's easy enough to calculate the next member in the sequence by using the divisor function \(\sigma_1 \). For example, the second term in the sequence starting with 138 is \(\sigma_1 (138) -138=150\). Many sequences lead to a prime number and then terminate because \(\sigma_1 (\text{prime number}) -\text{prime number}=1\) and \(\sigma_1(1) -1=0\). The sequence beginning with 138 (OEIS A008888) has 178 members and ends in 59, 1, 0. OEIS A008889 is really the same as OEIS A008888 except for the starting point (150 instead of 138)

Aliquot sequence OEIS A008890 is different however, and starts with 168 but the second term is 312 which is the fifth term in OEIS A008888. Aliquot sequence OEIS A074907 starts with 570 but after a few terms reaches 19434 which again is a term in the OEIS A08888 sequence.

Not all aliquot sequences end. To quote from Wikipedia:

There are a variety of ways in which an aliquot sequence might not terminate:

• A perfect number has a repeating aliquot sequence of period 1. The aliquot sequence of 6, for example, is 6, 6, 6, 6, ... 

• An amicable number has a repeating aliquot sequence of period 2. For instance, the aliquot sequence of 220 is 220, 284, 220, 284, ... 

• A sociable number has a repeating aliquot sequence of period 3 or greater. (Sometimes the term sociable number is used to encompass amicable numbers as well.) For instance, the aliquot sequence of 1264460 is 1264460, 1547860, 1727636, 1305184, 1264460, ... 

• Some numbers have an aliquot sequence which is eventually periodic, but the number itself is not perfect, amicable, or sociable. For instance, the aliquot sequence of 95 is 95, 25, 6, 6, 6, 6, ... . Numbers like 95 that are not perfect, but have an eventually repeating aliquot sequence of period 1 are called aspiring numbers (OEIS  A063769).

Numbers whose Aliquot sequence is not known to be finite or eventually periodic are:

276, 306, 396, 552, 564, 660, 696, 780, 828, 888, 966, 996, 1074, 1086, 1098, 1104, 1134, 1218, 1302, 1314, 1320, 1338, 1350, 1356, 1392, 1398, 1410, 1464, 1476, 1488, ... (sequence A131884 in the OEIS) 

ADDENDUM: 17th July 2020

Today, I turned 26038 days old and I revisited OEIS A008888 (Aliquot sequence starting at 138) because this number is a member of the sequence and appears very near the end. Here is the full sequence:

138, 150, 222, 234, 312, 528, 960, 2088, 3762, 5598, 6570, 10746, 13254, 13830, 19434, 20886, 21606, 25098, 26742, 26754, 40446, 63234, 77406, 110754, 171486, 253458, 295740, 647748, 1077612, 1467588, 1956812, 2109796, 1889486, 953914, 668966, 353578, 176792, 254128, 308832, 502104, 753216, 1240176, 2422288, 2697920, 3727264, 3655076, 2760844, 2100740, 2310856, 2455544, 3212776, 3751064, 3282196, 2723020, 3035684, 2299240, 2988440, 5297320, 8325080, 11222920, 15359480, 19199440, 28875608, 25266172, 19406148, 26552604, 40541052, 54202884, 72270540, 147793668, 228408732, 348957876, 508132204, 404465636, 303708376, 290504024, 312058216, 294959384, 290622016, 286081174, 151737434, 75868720, 108199856, 101437396, 76247552, 76099654, 42387146, 21679318, 12752594, 7278382, 3660794, 1855066, 927536, 932464, 1013592, 1546008, 2425752, 5084088, 8436192, 13709064, 20563656, 33082104, 57142536, 99483384, 245978376, 487384824, 745600776, 1118401224, 1677601896, 2538372504, 4119772776, 8030724504, 14097017496, 21148436904, 40381357656, 60572036544, 100039354704, 179931895322, 94685963278, 51399021218, 28358080762, 18046051430, 17396081338, 8698040672, 8426226964, 6319670230, 5422685354, 3217383766, 1739126474, 996366646, 636221402, 318217798, 195756362, 101900794, 54202694, 49799866, 24930374, 17971642, 11130830, 8904682, 4913018, 3126502, 1574810, 1473382, 736694, 541162, 312470, 249994, 127286, 69898, 34952, 34708, 26038, 13994, 7000, 11720, 14740, 19532, 16588, 18692, 14026, 7016, 6154, 3674, 2374, 1190, 1402, 704, 820, 944, 916, 694, 350, 394, 200, 265, 59, 1, 0.

This is not the last time I'll encounter OEIS A008888 because in a couple of years time, I'll meet 26742 and 26754 (assuming I'm still alive).

Building Brilliant Numbers

I shouldn't let the opportunity go by to make a note of today and tomorrow's numbers: 25094 and 25095. Both these numbers belong to the OEIS sequence A108770: numbers n such that \(n^2 + (n+1)^2 \) is a brilliant number. The sequence proceeds:

3, 10, 15, 20, 27, 37, 59, 92, 105, 120, 152, 155, 175, 190, 215, 219, 242, 245, 254, 255, 277, 300, 302, 307, 325, 337, 362, 365, 370, 402, 415, 614, 930, 944, 987, 1049, 1059, 1112, 1192, 1204, 1210, 1220, 1265, 1312, 1344, 1360, 1374, 1449, 1460, 1504, 1527, ...

As can be seen, such numbers are relatively common. 92 is given as an example \( 92^2 + 93^2 = 17113 = 109*157 \) as both of its factors have three digits. 25094 is also in this sequence because \( 25094^2+25095^2 = 1259467861 = 23873×52757 \) but so also is 25095 because \( 25095^2+25096^2 = 1259568241 = 29401×42841 \). These consecutive pairs are relatively rare and those listed are (254,255), (4099,4100), (11159,11160), (25094, 25095), (31754,31755) and (40189,40190) with the conjecture that there are infinitely many of such pairs.

It's a bit of a wait until the next pair (31754 and 31755) but hopefully I'll be around to greet them.

Triangulation Conjecture Disproved

This is a story from Quanta Magazine about a Romanian-American mathematician by the name of Ciprian Manolescu (Wikipedia link) who was born on December 24th 1978. He is now approaching his 39th birthday. The article appeared in January of 2015 but I only read about today via Flipboard. He managed to disprove the triangulation conjecture, which posits that all manifolds (abstract spaces) can be triangulated (one of the most famous problems in topology). He has "the sole distinction of writing three perfect papers at the International Mathematical Olympiad: Toronto, Canada (1995); Bombay, India (1996); Mar del Plata, Argentina (1997)" and "he was elected as a member of the 2017 class of Fellows of the American Mathematical Society for contributions to Floer homology and the topology of manifolds" (quoting from the Wikipedia article).

The article mentions invariants and uses the well known Euler Formula for polyhedra stating that the number of vertices V, faces F, and edges E in a convex 3-dimensional polyhedron, satisfies V + F - E = 2.

An “invariant” is a tool mathematicians use to compare spaces, or manifolds. One famous example is the Euler characteristic, shown here. To calculate it for any two-dimensional manifold, first carve the manifold into polygons (here we use triangles). Next, add the number of faces to the number of vertices and subtract the number of edges. Every sphere will have an Euler characteristic of 2, no matter how the manifold is carved up.

To quote from the article:

He created a new invariant, which he named “beta,” and used it to create a proof by contradiction. Here’s how it works: As we have seen, the triangulation conjecture is equivalent to asking whether there exists a homology 3-sphere with certain characteristics. One characteristic is that the sphere has to have a certain property — a Rokhlin invariant of 1. Manolescu showed that when a homology 3-sphere has a Rokhlin invariant of 1, the value of beta has to be odd. At the same time, other necessary characteristics of these homology 3-spheres require beta to be even. Since beta cannot be both even and odd at the same time, these particular homology 3-spheres do not exist. Thus, the triangulation conjecture is false.

Here is a link to Ciprian Manolescu's UCLA page.

Triangles with Optimal Dynamics

In my previous post, I explored the close-to-equilateral integer triangles and today I was reading another mathematical article about triangles, so-called triangles with optimal dynamics. I thought the topic worthy of a blog post. Here is a quote from the article:

When you set a ball in motion on a billiard table, it may seem as if anything is possible, but when a table has optimal dynamics, only two things truly are. The first is complete chaos, which is to say that the ball’s path will cover the entire table as time wears on. The second is periodicity — a repeating path like a ball pinging back and forth between two sides.

In tables without optimal dynamics, a wider range of possibilities exists, which makes the full analysis of all possible paths impossible: A ball could end up bouncing chaotically in one part of the table forever, never retracing its path, but also never covering the whole table.

What mathematicians do know is that there are at least eight kinds of triangles with optimal dynamics; the first was discovered in 1989 and the last in 2013. Whether there are more is anyone’s guess. 

 Let's look at each in turn:

• 1 : 1 : n means isosceles triangles with angles 10°, 10°, 160° or 20°, 20°, 140° etc.
  
• 1 : 2 : n means triangles with angles 10°, 20°, 150° or 20°, 40°, 120° etc.
  
• 3 : 4 : 5 represents a triangle with angles of 45°, 60° and 75°
  
• 2 : 3 : 4 represents a triangle with angles of 40°, 60° and 80°
  
• 3 : 5 : 7 represents a triangle with angles of 36°, 60° and 84°
  
• 1 : 4 : 7 represents a triangle with angles of 15°, 60° and 105°
  
• 2 : (n-2) : n represents a right angled triangles with angles of 2°, 88° or 90°, 4°, 86° or 6°, 84°, 90° or 10°, 80°, 90° or 12°, 78°, 90° or 18°, 72°, 90° or 20°, 70°, 90° or 30°, 60°, 90° or 36°, 54°, 90°
  
• 2 : n : n represents triangles like 2°, 89°,89° or 4°, 88°, 88° or 6°, 86°, 86° etc.

The article goes on to identify two quadrilaterals with optimal dynamics (shown below):

25080: A Number Of Interest

The number of OEIS entries for numbers in the region 25000 can be quite few. For example, 25079 has three entries, 25078 has six, 25077 has five and so on. Thus it was surprising to note that 25080 has an impressive 43 entries. It helps that it's highly factorisable: 2 x 2 x 2 × 3 × 5 × 11 × 19. Let's look at one of these entries.

OEIS A102341: areas of 'close-to-equilateral' integer triangles. To quote from OEIS:

A close-to-equilateral integer triangle is defined to be a triangle with integer sides and integer area such that the largest and smallest sides differ in length by unity. The first five close-to-equilateral integer triangles have sides (5, 5, 6), (17, 17, 16), (65, 65, 66), (241, 241, 240) and (901, 901, 902).

The corresponding sequence of areas for these triangles is 12, 120, 1848, 25080, 351780. So the 'close-to-equilateral' integer triangle with sides 241, 241 and 240 has an integer area of 25080. Here is the image that I posted to my Twitter account, taken from WolframAlpha:

Here is further detail from the OEIS entry:

Heron's formula: a triangle with side lengths \( (x,y,z)  \) has area \(A = \sqrt {s*(s-x)*(s-y)*(s-z)} \) where \(s = (x+y+z)/\)2. For this sequence we assume integer side-lengths \(x = y = z \pm 1 \). Then for \(A \) to also be an integer, \( x+y+z \) must be even, so we can assume \( z = 2k \) for some positive integer \( k \). Now \( s = (x+y+z)/2 = 3k \pm 1 \) and \(A = \sqrt{(3*k \pm 1)*k*k*(k \pm 1)} = k*\sqrt{3*k^2 \pm 4*k + 1} \). To determine when this is an integer, set \( 3*k^2 \pm 4*k + 1 = d^2 \). If we multiply both sides by 3, it is easier to complete the square: \((3*k \pm 2)^2 - 1 = 3*d^2 \). Now we are looking for solutions to the Pell equation \( c^2 - 3*d^2 = 1 \) with \( c = 3*k \pm 2 \), for which there are infinitely many solutions: use the upper principal convergents of the continued fraction expansion of \( \sqrt{3} \). 

 Here are links supplied in the OEIS entry:

•   Danny Rorabaugh, Table of n, a(n) for n = 1..875
•   Steven Dutch, Almost 30-60 Triples
•   ProjectEuler, Problem 94: Almost equilateral triangles
•   Eric Weisstein's World of Mathematics, Heronian Triangle and Pell Equation