Collatz Trajectory Crossing Records

One of the properties associated with my diurnal age's number (27681) is that it is a member of OEIS A319738:

 A319738: Numbers whose Collatz trajectories cross their initial values a record number of times.

The initial members of the sequence are: 1, 3, 6, 9, 14, 18, 33, 54, 97, 129, 194, 257, 294, 313, 342, 353, 398, 417, 470, 626, 9225, 13739, 14473, 19297, 27681, 38881 (permalink)

Table 1 shows the progressive records (permalink):

Table 1

Figure 1 shows the Collatz trajectory of 27681 in which there are 61 crossings (permalink):

Figure 1

These record crossings are rather rare and I don't know if I'll experience the next one. I'll be 38881 days old which I'm unlikely to reach. The reason that I've never encountered these record number of crossings before is that the previous one occurred when I was 19297 days old, long before I started this daily number analysis.

Vampire Numbers

There is more than one type of vampire number but the first type that I'll deal with in this post belongs to OEIS A014575:

A014575: Vampire numbers (definition 2): numbers \(n\) with an even number of digits which have a factorization \(n = i \times j\) where \(i\) and \(j\) have the same number of digits and the multiset of the digits of \(n\) coincides with the multiset of the digits of \(i\) and \(j\).

Examples are \(1260=21 \times 60\) and \(939658=953 \times 986\). The two relevant divisors of a vampire number are called its fangs and the numbers we are dealing with here have two fangs.

Up to one million, the vampire numbers are (permalink):

1260, 1395, 1435, 1530, 1827, 2187, 6880, 102510, 104260, 105210, 105264, 105750, 108135, 110758, 115672, 116725, 117067, 118440, 120600, 123354, 124483, 125248, 125433, 125460, 125460, 125500, 126027, 126846, 129640, 129775, 131242, 132430, 133245, 134725, 135828, 135837, 136525, 136948, 140350, 145314, 146137, 146952, 150300, 152608, 152685, 153436, 156240, 156289, 156915, 162976, 163944, 172822, 173250, 174370, 175329, 180225, 180297, 182250, 182650, 186624, 190260, 192150, 193257, 193945, 197725, 201852, 205785, 211896, 213466, 215860, 216733, 217638, 218488, 226498, 226872, 229648, 233896, 241564, 245182, 251896, 253750, 254740, 260338, 262984, 263074, 284598, 284760, 286416, 296320, 304717, 312475, 312975, 315594, 315900, 319059, 319536, 326452, 329346, 329656, 336550, 336960, 338296, 341653, 346968, 361989, 362992, 365638, 368550, 369189, 371893, 378400, 378418, 378450, 384912, 386415, 392566, 404968, 414895, 416650, 416988, 428980, 429664, 447916, 456840, 457600, 458640, 475380, 486720, 489159, 489955, 498550, 516879, 529672, 536539, 538650, 559188, 567648, 568750, 629680, 638950, 673920, 679500, 729688, 736695, 738468, 769792, 789250, 789525, 792585, 794088, 809919, 809964, 815958, 829696, 841995, 939658

As can be seen, \(1260\) is the first four digit vampire number and \(6880=80 \times 86\) is the last. The first six digit number is \(102510=201 \times 510\) and the last is \(939658\). Of the 156 numbers listed above, it can be seen that \(125460\) appears twice and this is because it has two representations viz. \(204 \times 615\) and \(246 \times 510\). 

\(13078260\) is an example of a vampire number that has three representations:$$ \begin{align} 13078260 &=1620 \times 8073\\&=1863 \times 7020 \\&=2070 \times 6318 \end{align}$$Wolfram Mathworld has examples of numbers that can be represented in four and five different ways.

Another type of vampire number is listed in OEIS A020342:

A020342: Vampire numbers: (definition 1): \(n\) has a nontrivial factorization using \(n\)'s digits. Nontrivial means that there must be at least two factors.

An example is \(126 = 6 \times 21\) and \(39784=8 \times 4973\). The initial members of this sequence are up to 40000 (permalink):

126, 153, 688, 1206, 1255, 1260, 1260, 1395, 1435, 1503, 1530, 1530, 1827, 2187, 3159, 3784, 6880, 6880, 10251, 10255, 10426, 10521, 10525, 10575, 11259, 11844, 11848, 12006, 12060, 12060, 12384, 12505, 12546, 12550, 12550, 12595, 12600, 12600, 12600, 12762, 12843, 12955, 12964, 13243, 13545, 13950, 13950, 14035, 14350, 14350, 15003, 15030, 15030, 15246, 15300, 15300, 15300, 15435, 15624, 15795, 16272, 17325, 17428, 17437, 17482, 18225, 18265, 18270, 18270, 19026, 19215, 21375, 21586, 21753, 21870, 21870, 25105, 25375, 25474, 25510, 28476, 29632, 31509, 31590, 31590, 33655, 33696, 36855, 37840, 37840, 37845, 39784

There are 712 numbers in the range up to one million, with some repeated numbers of course. All of the members of OEIS A014575 are in this sequence but the conditions (that the number must have an even number of digits and have its two divisors of equal length and not both ending in zero) have been relaxed. Consequently, any number \(n\) in the sequence will also have \(n \times 10\) in the sequence.

Aliquant Parts

I came across a mathematical term today that I hadn't heard of before. The term is "aliquant" defined as follows by contrasting it to similar sounding "aliquot" (source):

Webster defines 'aliquot' as something that contained an exact number of times in something else or to divide into equal parts.

Notice the word "equal". An example being 5 is an aliquot part of 15.

The term 'aliquant', however, is slightly different. Defined as being a part of a number or quantity, but not dividing it without leaving a remainder. An example being 5 is an aliquant part of 16. 

The term occurred in the following context:

 A098743: number of partitions of \(n\) into aliquant parts (i.e., parts that do not divide \(n\)). 

The initial members of the sequence are:

1, 0, 0, 0, 0, 1, 0, 3, 1, 3, 3, 13, 1, 23, 10, 11, 9, 65, 8, 104, 14, 56, 66, 252, 10, 245, 147, 206, 77, 846, 35, 1237, 166, 649, 634, 1078, 60, 3659, 1244, 1850, 236, 7244, 299, 10086, 1228, 1858, 4421, 19195, 243, 17660, 3244, 12268, 4039, 48341, 1819, 27675

The last member shown above, 27675, is my diurnal age today and corresponds to \(n=55\). To find the aliquant parts, simply remove the divisors of the number. For example, 55 has divisors of 1, 5, 11 and 55 and so the number of aliquant parts is 51. The list is as follows:

2, 3, 4, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54

Note that these are different to the numbers that contribute to the total of the totient of a number, where the numbers are coprime. The totient for 55 is 40, made up of the following numbers.

1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 16, 17, 18, 19, 21, 23, 24, 26, 27, 28, 29, 31, 32, 34, 36, 37, 38, 39, 41, 42, 43, 46, 47, 48, 49, 51, 52, 53, 54

Welcoming in 2025

\(2025\) is a very "squarish" number as the following properties will illustrate:

• \(2025 =45^2=27^2+36^2\)
• \(2 + 0 + 2 + 5 = 9 = 3^2\)
• It is divisible by the square of the sum of its digits:
  \( \dfrac{2025}{81}=25\)
• Omitting the zero still leaves a square \(225=15^2\)
• Omitting the first two digits still leaves a square \(25=5^2\)
• Adding \(1\) to the first digit gives the square \(3025 = 55^2\)
• Adding \(1\) to each digit gives the square \(3136=56^2\)
• Adding \(4\) at the front gives a square \(42025 =205^2\)
• \(2025\) is the smallest square that can be formed from \(20\) by adding one or more digits
• Square that can be seen on a digital clock as in \(20:25\)
• Written as "twenty twenty five" it has \(16=4^2\) letters
• Can be written as a sum of three distinct squares in \(9=3^2\) different ways e.g. \(4^2+ 28^2+ 35^2\) ... permalink
• It is the sum of the first nine numbers squared: \( (1+2+ \dots +8+9)^2=2025\)
• Deleting a zero from its cube (\(8303765625\)) gives \(833765625=28875^2\)
• Imagine writing down the number \(1\) once, the number \(2\) twice, the number \(3\) three times, and so on up to the number \(45\) forty-five times, like this:$$12233344445555 \dots 454545$$The total number of digits is \(2025\), which is the square of \(45\). This coincidence does not occur for any other number greater than \(1\).
• The sum of entries (in red, below) of a \(9 \times 9\) multiplication table is \(2025\):

• The determinant of the \(4 \times 4\) matrix whose rows are cyclic permutations of the first \(4\) composite numbers is equal to \(-2025\).

A Bit About Betting

I got to thinking about the way that the odds of a horse winning are expressed in betting circles and I sought the help of Gemini 2.0 Flash Experimental to explain it. Here is what it came up with.

Question: If probability of an event happening is given as 2 to 1 on, what does that mean?

When someone says the probability of an event is "2 to 1 on," they're expressing odds in fractional form, specifically "odds on." This means the event is considered more likely to happen than not.   

Here's how to break it down:

2 to 1: This represents the ratio of the likelihood of the event happening versus it not happening. In this case, it means for every 2 times the event is expected to occur, it is expected not to occur 1 time.

"On": This indicates that the first number in the ratio (2 in this case) is larger than the second number (1). This signifies that the event is favored or has a greater than 50% chance of happening.

To convert "2 to 1 on" to a probability:

Add the two numbers in the ratio: 2 + 1 = 3

Divide the first number by the total: 2 / 3 = 0.666...

Therefore, "2 to 1 on" translates to approximately a 66.7% probability or two-thirds chance of the event happening.

Example:

If you're told the odds of a horse winning a race are "2 to 1 on," it implies that out of every three similar races, the horse is expected to win two and lose one.

Key takeaway: "Odds on" always indicate a greater than 50% chance of an event occurring, with the specific probability determined by the ratio provided.

I then asked the following question:

Question: What's an example of a bet that isn't odds on?

You're right to ask for clarification! It's important to understand the difference. A bet that isn't "odds on" means the event is considered less likely to happen than not. This is often referred to as "odds against."

Here's how it works:

The ratio: In "odds against," the second number in the ratio is larger than the first. For example, "3 to 1 against" or "5 to 2 against."

Meaning: This signifies that the event is less favored or has a less than 50% chance of happening.

Example:

Let's say the odds of a particular football team winning their next match are "3 to 1 against."

This means that for every 3 times the team is expected not to win, they are expected to win only 1 time.

To convert this to a probability:

Add the two numbers: 3 + 1 = 4

Divide the second number by the total: 1 / 4 = 0.25

So, "3 to 1 against" translates to a 25% probability of the team winning.

Another example:

If the odds of rolling a 6 on a standard six-sided die are "5 to 1 against," this means that for every 5 times you're expected not to roll a 6, you're expected to roll a 6 only once. This aligns with the actual probability of 1/6 (approximately 16.7%).

In summary:

Odds on: The first number in the ratio is larger (e.g., 2 to 1 on). Indicates a greater than 50% chance.

Odds against: The second number in the ratio is larger (e.g., 3 to 1 against).1 Indicates a less than 50% chance.   

Understanding this distinction is crucial for interpreting betting odds and evaluating the likelihood of different outcomes.

Gapful Numbers

 Numbers Aplenty has this to say about gapful numbers:

L. Colucci calls a number \(n\) of at least 3 digits a gapful number if \(n\)  is divisible by the number formed by the first and last digit of \(n\). For example, 583 is gapful because it is divisible by 53. About the 7.46% of the numbers are gapful.

The smallest Pythagorean triple made of gapful numbers is (120, 160, 200) and the first such primitive triple is (3159, 29440, 29609).

The concept can be extended to other bases. For example, the number 240 is gapful in all the bases from 2 to 10 and the number 54288 is gapful in all the bases from 2 to 16.

The first gapful numbers are 100, 105, 108, 110, 120, 121, 130, 132, 135, 140, 143, 150, 154, 160, 165, 170, 176, 180, 187, 190, 192, 195, 198, 200, 220, 225, 231, 240, 242, 253  

Figure 1 shows the representation of \(54288 = 2^4 \times 3^2 \times 13 \times 29\) in bases from 2 to 16:

Figure 1

Looking at the first and last digits in these different representations we see that:

• Base 2 --> 10 which is 2

• Base 3 --> 20 which is 6

• Base 4 --> 30 which is 12

• Base 5 --> 33 which is 18

• Base 6 --> 10 which is 6

• Base 7 --> 33 which is 24

• Base 8 --> 10 which is 8

• Base 9 --> 80 which is 72

• Base 10 --> 58 which is 58

• Base 11 --> 33 which is 36

• Base 12 --> 20 which is 24

• Base 13 --> 10 which is 13

• Base 14 --> 1a which is 24

• Base 15 --> 13 which is 18

• Base 16 --> d0 which is 13 x 16 = 208

All these numbers (2, 6, 8, 12 etc.) are divisors of 54288 and this why the number is so special.

One way to make these gapful numbers less numerous is to impose an additional criterion. For example, let's require that the sum of the number's digits is equal to the concatenation of the first and last digits. In the range between 1 and 40000, this reduces the numbers satisfying all criteria to 359. The numbers are (permalink):

190, 192, 195, 198, 1090, 1092, 1095, 1098, 1180, 1183, 1185, 1188, 1270, 1272, 1274, 1275, 1278, 1360, 1365, 1368, 1450, 1452, 1455, 1456, 1458, 1540, 1545, 1547, 1548, 1630, 1632, 1635, 1638, 1720, 1725, 1728, 1729, 1810, 1812, 1815, 1818, 1900, 1904, 1905, 1908, 2992, 2997, 10090, 10092, 10094, 10095, 10096, 10098, 10180, 10185, 10188, 10270, 10272, 10275, 10278, 10279, 10360, 10365, 10368, 10450, 10452, 10455, 10458, 10540, 10543, 10545, 10548, 10630, 10632, 10635, 10638, 10720, 10724, 10725, 10727, 10728, 10810, 10812, 10815, 10816, 10818, 10900, 10901, 10905, 10908, 11080, 11085, 11088, 11170, 11172, 11175, 11178, 11260, 11265, 11268, 11350, 11352, 11354, 11355, 11358, 11440, 11445, 11448, 11530, 11532, 11535, 11536, 11538, 11620, 11625, 11628, 11710, 11712, 11713, 11715, 11718, 11800, 11805, 11808, 12070, 12072, 12075, 12078, 12160, 12165, 12168, 12250, 12252, 12255, 12256, 12257, 12258, 12340, 12345, 12348, 12430, 12432, 12435, 12438, 12520, 12525, 12528, 12610, 12612, 12614, 12615, 12618, 12700, 12705, 12708, 13060, 13065, 13068, 13150, 13152, 13155, 13158, 13240, 13244, 13245, 13248, 13330, 13332, 13335, 13338, 13420, 13425, 13428, 13510, 13512, 13515, 13518, 13600, 13605, 13608, 14050, 14052, 14053, 14055, 14058, 14140, 14145, 14148, 14230, 14232, 14235, 14238, 14320, 14325, 14328, 14410, 14412, 14415, 14416, 14418, 14500, 14504, 14505, 14508, 15040, 15045, 15048, 15130, 15132, 15134, 15135, 15136, 15138, 15220, 15223, 15225, 15228, 15310, 15312, 15315, 15317, 15318, 15400, 15405, 15408, 15409, 16030, 16032, 16035, 16038, 16120, 16125, 16128, 16210, 16212, 16215, 16218, 16300, 16305, 16308, 17020, 17024, 17025, 17028, 17110, 17112, 17115, 17118, 17119, 17200, 17205, 17208, 18010, 18012, 18015, 18016, 18018, 18100, 18105, 18108, 19000, 19005, 19008, 21897, 21980, 21984, 22792, 22793, 22880, 22975, 22977, 23780, 23782, 23784, 23787, 23875, 23960, 23961, 23968, 24591, 24596, 24597, 24680, 24772, 24775, 24860, 24863, 24864, 24867, 25580, 25584, 25675, 25677, 25760, 25762, 25766, 25851, 25940, 25944, 25947, 26480, 26481, 26487, 26488, 26575, 26660, 26664, 26752, 26757, 26840, 26933, 26936, 27297, 27380, 27384, 27475, 27560, 27567, 27740, 27741, 27742, 27744, 27748, 27837, 27920, 27925, 28280, 28371, 28375, 28377, 28460, 28464, 28640, 28647, 28732, 28739, 28820, 28824, 28825, 28917, 29180, 29184, 29187, 29275, 29276, 29360, 29457, 29540, 29544, 29631, 29720, 29722, 29725, 29727, 29900, 29904, 39990, 39996, 39997

For example, 39997 has a first and last digit concatenation of 37 with a digit of 37 and a factoriation of 23 x 37 x 47. 

If instead, we require the product of the number's digit instead then only five numbers satisfy: 135, 11232, 12132, 12312, 13212 (permalink). For example, 13212 = 2^2 x 3^2 x 367 has a digit product of 12 which is a divisor of the number and 12 is also the concatenation of the first and last digits of the number.

Another criterion that could be applied is to require that the first and last digits be the same. In this case 227 numbers satisfy in the range up to 40000. They are (permalink):

121, 242, 363, 484, 1001, 1111, 1221, 1331, 1441, 1551, 1661, 1771, 1881, 1991, 2002, 2112, 2222, 2332, 2442, 2552, 2662, 2772, 2882, 2992, 3003, 3333, 3663, 3993, 4004, 4224, 4444, 4664, 4884, 5005, 5115, 5225, 5335, 5445, 5555, 5665, 5775, 5885, 5995, 6006, 6336, 6666, 6996, 7007, 7777, 8008, 8448, 8888, 9009, 9999, 10021, 10131, 10241, 10351, 10461, 10571, 10681, 10791, 10901, 11011, 11121, 11231, 11341, 11451, 11561, 11671, 11781, 11891, 12001, 12111, 12221, 12331, 12441, 12551, 12661, 12771, 12881, 12991, 13101, 13211, 13321, 13431, 13541, 13651, 13761, 13871, 13981, 14091, 14201, 14311, 14421, 14531, 14641, 14751, 14861, 14971, 15081, 15191, 15301, 15411, 15521, 15631, 15741, 15851, 15961, 16071, 16181, 16291, 16401, 16511, 16621, 16731, 16841, 16951, 17061, 17171, 17281, 17391, 17501, 17611, 17721, 17831, 17941, 18051, 18161, 18271, 18381, 18491, 18601, 18711, 18821, 18931, 19041, 19151, 19261, 19371, 19481, 19591, 19701, 19811, 19921, 20042, 20152, 20262, 20372, 20482, 20592, 20702, 20812, 20922, 21032, 21142, 21252, 21362, 21472, 21582, 21692, 21802, 21912, 22022, 22132, 22242, 22352, 22462, 22572, 22682, 22792, 22902, 23012, 23122, 23232, 23342, 23452, 23562, 23672, 23782, 23892, 24002, 24112, 24222, 24332, 24442, 24552, 24662, 24772, 24882, 24992, 25102, 25212, 25322, 25432, 25542, 25652, 25762, 25872, 25982, 26092, 26202, 26312, 26422, 26532, 26642, 26752, 26862, 26972, 27082, 27192, 27302, 27412, 27522, 27632, 27742, 27852, 27962, 28072, 28182, 28292, 28402, 28512, 28622, 28732, 28842, 28952, 29062, 29172, 29282, 29392, 29502, 29612, 29722, 29832, 29942, 30063, 30393, 30723, 31053, 31383, 31713, 32043, 32373, 32703, 33033, 33363, 33693, 34023, 34353, 34683, 35013, 35343, 35673, 36003, 36333, 36663, 36993, 37323, 37653, 37983, 38313, 38643, 38973, 39303, 39633, 39963

For example, 39963 = 3 x 7 x 11 x 173 has first and last digits the same and 33 divides the number.

Jordan-Polya Numbers

The number associated with my diurnal age today, 27648, is a Jordan-Polya number which Numbers Aplenty defines as a number that can be written as the product of factorial numbers. In the case of 27648 we have:$$\begin{align} 27648 &=2^{10} \times 3^3\\&=(4!)^3 \times 2! \end{align} $$There aren't that many of them in the range up to 40,000. Here are the initial numbers:

1, 2, 4, 6, 8, 12, 16, 24, 32, 36, 48, 64, 72, 96, 120, 128, 144, 192, 216, 240, 256, 288, 384, 432, 480, 512, 576, 720, 768, 864, 960, 1024, 1152, 1296, 1440, 1536, 1728, 1920, 2048, 2304, 2592, 2880, 3072, 3456, 3840, 4096, 4320, 4608, 5040, 5184, 5760, 6144, 6912, 7680, 7776, 8192, 8640, 9216, 10080, 10368, 11520, 12288, 13824, 14400, 15360, 15552, 16384, 17280, 18432, 20160, 20736, 23040, 24576, 25920, 27648, 28800, 30240, 30720, 31104, 32768, 34560, 36864

Here is what Wikipedia had to say about them:

In mathematics, the Jordan–Pólya numbers are the numbers that can be obtained by multiplying together one or more factorials, not required to be distinct from each other. For instance, 480 is a Jordan–Pólya number because:$$480 =2! \times 2! \times 5!$$Every tree has a number of symmetries that is a Jordan–Pólya number, and every Jordan–Pólya number arises in this way as the order of an automorphism group of a tree. These numbers are named after Camille Jordan and George Pólya, who both wrote about them in the context of symmetries of trees.

These numbers grow more quickly than polynomials but more slowly than exponentials. As well as in the symmetries of trees, they arise as the numbers of transitive orientations of comparability graphs and in the problem of finding factorials that can be represented as products of smaller factorials.

A tree that (as an abstract graph) has 480 symmetries (automorphisms). There are 2 ways of permuting the two children of the upper left vertex, 2 ways of permuting the two children of the upper middle vertex, and 5! = 120 ways of permuting the five children of the upper right vertex, for 2 x 2 x 120 = 480 symmetries altogether.

George Pólya
13 December 1887 - 7 September 1985

George Pólya was one of the greatest mathematicians of the 20th century and there is a brief biography of his life to be found here. To quote from this article:

With no hesitation, George Pólya is my personal hero as a mathematician. ... [he] is not only a distinguished gentleman but a most kind and gentle man: his ebullient enthusiasm, the twinkle in his eye, his tremendous curiosity, his generosity with his time, his spry energetic walk, his warm genuine friendliness, his welcoming visitors into his home and showing them his pictures of great mathematicians he has known - these are all components of his happy personality. As a mathematician, his depth, speed, brilliance, versatility, power and universality are all inspiring. Would that there were a way of teaching and learning these traits.

Camille Jordan
5th January 1838 - 22nd January 1922

Camille Jordan made most of his mathematical contributions in the 19th century and a brief biography can be found here. It can be noted that, quoting from the article:

The Journal de Mathématiques Pure et Appliquées was a leading mathematical journal and played a very significant part in the development of mathematics throughout the 19th century. It was usually known as the Journal de Liouville since Liouville had founded the journal in 1836. Liouville died in 1882 and in 1885 Jordan became editor of the Journal, a role he kept for over 35 years until his death.