Another Weird Day

It's been 560 days since my last $\mathbf{\text{weird}}$ day. Today I turned 27790 days old and once again it's weird. I've written about this type of number before in blog posts titled Weird Numbers on August 4th 2019 and also in an earlier post titled Zumkellar Numbers, Half Zumkellar Numbers and Pseudoperfect Numbers on November 22nd 2018.

Most abundant numbers are pseudoperfect meaning that a subset of their proper divisors can be chosen so that their sum is equal to the number. If the sum of all the proper divisors of a number equals the number itself, then the number is said to be perfect. The first few perfect numbers are 6, 28, 496, 8128 and 33550336. An example of an abundant number that is pseudoperfect is 24 with proper divisors of 1, 2, 3, 4, 6, 8 and 12. Adding these gives 36 so the number is clearly abundant. However, if we add the proper divisors without including 12, we reach 24 and so the number is pseudoperfect.

There are two types of weird numbers: primitive and non-primitive. A non-primitive weird number is a number that is a multiple of a weird number. The primitive weird numbers up to 40000 are 70, 836, 4030, 5830, 7192, 7912, 9272, 10792 and 17272. Here is a list of all weird numbers (primitive and non-primitive) up to 40000:

70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, 15610, 15890, 16030, 16310, 16730, 16870, 17272, 17570, 17990, 18410, 18830, 18970, 19390, 19670, 19810, 20510, 21490, 21770, 21910, 22190, 23170, 23590, 24290, 24430, 24710, 25130, 25690, 26110, 26530, 26810, 27230, 27790, 28070, 28630, 29330, 29470, 30170, 30310, 30730, 31010, 31430, 31990, 32270, 32410, 32690, 33530, 34090, 34370, 34930, 35210, 35630, 36470, 36610, 37870, 38290, 38990, 39410, 39830, 39970

All the numbers above with the exception of the primitive weird numbers in red are multiples of 70. Any weird number multiplied by a prime number that is greater than the sum of the divisors of the numbers is itself weird. Thus 70 has a sum of divisors of 144 and the first prime above this is 149. 70 x 149 produces the weird number 10430.

As can be seen, my next weird number (28070) is 280 days away and after that there is a gap of 560 days to my next weird number (28630). The gaps of course are determined by the distance between successive prime numbers. Thus:$$\begin{array}{rl}27790& =70\times 397\\ 28070& =70\times 401\\ 28630& =70\times 409\end{array}$$Here is a fuller list of primitive weird numbers (link):

70, 836, 4030, 5830, 7192, 7912, 9272, 10792, 17272, 45356, 73616, 83312, 91388, 113072, 243892, 254012, 338572, 343876, 388076, 519712, 539744, 555616, 682592, 786208, 1188256, 1229152, 1713592, 1901728, 2081824, 2189024, 3963968, 4128448

While this post is a little repetitive, I feel it's important to acknowledge the occurrence of weird numbers when they occur because of their rarity.

A Variation on the Descent to Zero

In my post on the 23rd of April 2025 titled Descent to Zero, I considered the smallest numbers that take a certain number of steps to reach 0 under "$k\to$ max product of two numbers whose concatenation is $k$". What happens if we change this slightly so that the rule is now "$k\to$ max product of two $\mathbf{\text{prime}}$ numbers whose concatenation is $k$".

This is highly restrictive because only numbers that can be split into a pair of prime numbers in one or more ways are eligible for consideration. For example, 246 is dismissed but 235 is eligible for consideration because it can be split into 23 x 5. Moreover, the process of splitting into primes needs to continue until zero is reached if the number is to be a candidate for the smallest number. Here are the numbers that require from 1 to 9 steps to reach zero:$$1,22,55,115,235,475,3389,13457,35743$$The breakdown is as follows:

• Descent of 9 steps to zero: 35743 --> 17229, 3893, 1167, 737, 511, 55, 25, 10, 0
• Descent of 8 steps to zero: 13457 --> 5941, 4705, 235, 115, 55, 25, 10, 0
• Descent of 7 steps to zero: 3389 --> 1167, 737, 511, 55, 25, 10, 0
• Descent of 6 steps to zero: 475 --> 235, 115, 55, 25, 10, 0
• Descent of 5 steps to zero: 235 --> 115, 55, 25, 10, 0
• Descent of 4 steps to zero: 115 --> 55, 25, 10, 0
• Descent of 3 steps to zero: 55 --> 25, 10, 0
• Descent of 2 steps to zero: 22 --> 4, 0
• Descent of 1 step to zero: 1 --> 0

The convention is that single digits or 10 get mapped to zero. Let's look at how 35743 reaches zero:$$\begin{array}{rl}35743& \to 3\times 5743=17229\\ 17229& \to 17\times 229=5743\\ 5743& \to 5\times 743=3893\\ 3893& \to 389\times 3=1167\\ 1167& \to 11\times 67=737\\ 737& \to 73\times 7=511\\ 511& \to 5\times 11=55\\ 55& \to 5\times 5=25\\ 25& \to 2\times 5=10\\ 10& \to 0\end{array}$$Note that other prime number products are possible. For example 5743 could be split into 57 x 43 but this is smaller than 5 x 5743. Similarly, 737 could be split into 7 x 37 but again this is smaller than 73 x 7.

There are in fact only 127 numbers in the range from 11 to 40000 that can be reduced down to 10 or a single digit. They are (permalink) with record breakers shown in red:

22, 23, 25, 32, 33, 52, 55, 112, 113, 115, 202, 203, 205, 211, 235, 297, 302, 303, 311, 415, 475, 502, 505, 511, 523, 541, 547, 583, 729, 737, 773, 835, 1012, 1013, 1015, 1102, 1103, 1105, 1153, 1167, 1512, 1675, 2002, 2003, 2005, 2011, 2101, 2151, 2251, 2305, 2512, 3002, 3003, 3011, 3101, 3389, 3893, 4015, 4105, 4437, 4615, 4705, 5002, 5005, 5011, 5023, 5041, 5047, 5083, 5101, 5167, 5401, 5461, 5821, 5941, 6711, 7029, 7073, 7171, 7443, 8215, 8305, 9415, 10102, 10103, 10105, 10171, 11002, 11003, 11005, 11053, 11067, 11191, 11491, 11643, 12743, 13457, 14537, 15102, 16705, 17229, 19111, 20002, 20003, 20005, 20011, 20101, 20151, 20251, 23005, 24183, 25051, 25102, 25501, 29659, 30002, 30003, 30011, 30101, 30389, 31971, 32237, 33881, 35743, 36437, 38813, 38903

This permalink will allow you to enter any of the above numbers and receive as output the descent of the number to 10 or a single digit. For example, entering the number 38903 produces the following output:

Starting with: 38903

Dividing 38903 into prime parts and multiplying gives: 1167

Dividing 1167 into prime parts and multiplying gives: 737

Dividing 737 into prime parts and multiplying gives: 511

Dividing 511 into prime parts and multiplying gives: 55

Dividing 55 into prime parts and multiplying gives: 25

Dividing 25 into prime parts and multiplying gives: 10

Reached: 10

ABA Numbers

The number (27783) associated with my diurnal age today has the property that it can be expressed as:$$27783=3\times {21}^3$$Numbers like this are called ABA numbers because they can be expressed in the form:$${\text{AB}}^{\text{A}}\text{ for A, B > 1}$$Up to 40,000, the ABA numbers are as follows (link):

8, 18, 24, 32, 50, 64, 72, 81, 98, 128, 160, 162, 192, 200, 242, 288, 324, 338, 375, 384, 392, 450, 512, 578, 648, 722, 800, 882, 896, 968, 1024, 1029, 1058, 1152, 1215, 1250, 1352, 1458, 1536, 1568, 1682, 1800, 1922, 2048, 2178, 2187, 2312, 2450, 2500, 2592, 2738, 2888, 3000, 3042, 3200, 3362, 3528, 3698, 3872, 3993, 4050, 4232, 4374, 4418, 4608, 4802, 5000, 5120, 5184, 5202, 5408, 5618, 5832, 6050, 6272, 6498, 6591, 6728, 6962, 7200, 7442, 7688, 7938, 8192, 8232, 8450, 8712, 8978, 9248, 9522, 9604, 9800, 10082, 10125, 10240, 10368, 10658, 10952, 11250, 11552, 11858, 12168, 12288, 12482, 12800, 13122, 13448, 13778, 14112, 14450, 14739, 14792, 15138, 15309, 15488, 15625, 15842, 16200, 16384, 16562, 16928, 17298, 17496, 17672, 18050, 18432, 18818, 19208, 19602, 20000, 20402, 20577, 20808, 21218, 21632, 22050, 22472, 22528, 22898, 23328, 23762, 24000, 24200, 24576, 24642, 25088, 25538, 25992, 26244, 26450, 26912, 27378, 27783, 27848, 28322, 28800, 29282, 29768, 30258, 30752, 31250, 31752, 31944, 32258, 32768, 33282, 33800, 34322, 34848, 35378, 35912, 36450, 36501, 36992, 37538, 38088, 38642, 38880, 39200, 39762, 40000

648 is the first ABA number with two representations:$$648=2\times {18}^2=3\times 6^3$$The smallest number with three such representations is 344373768:$$344373768=8\times 9^8=3\times {486}^3=2\times {13122}^2$$The smallest Pythagorean triples made of ABA numbers are:$$(98304,131072,163840)\text{ and }(229376,786432,819200)$$which correspond to:$$(3\times {32}^3,2\times {256}^2,5\times 8^5)\text{ and }(14\times 2^{14},3\times {64}^3,2\times {640}^2)$$

Some Special Sphenic Numbers

Today I turned 27782 days old and one of the properties of 27782 is that it is sphenic since:$$27782=2\times 29\times 479$$However, looking at the digital roots of the number and its factor we notice an interesting fact:$$\underset{8}{\underbrace{27782}}=\underset{2}{\underbrace{2}}\times \underset{2}{\underbrace{29}}\times \underset{2}{\underbrace{479}}$$The respective digital roots are shown under the number and its factors and we see that the digital roots of the factors (2) multiply together to give the digital root of the number (8). The only other way this can occur is if all the digital roots are 1.

In the range up to 100,000 there are 230 sphenic numbers with the property that the digital roots of the factors multiply to give the digital root of the number. These numbers are (permalink):

638, 1034, 1826, 2222, 2726, 3014, 3806, 4202, 4814, 4994, 5786, 5858, 6182, 6974, 7766, 7802, 7946, 8558, 9494, 9746, 10034, 10142, 10538, 11078, 12518, 12878, 12914, 13166, 14102, 14498, 14894, 14993, 15254, 16262, 16298, 16766, 17954, 18062, 18386, 18458, 18854, 20042, 20438, 20474, 20834, 21338, 21626, 22418, 22562, 22742, 24002, 24398, 24722, 25586, 25694, 25982, 26414, 26477, 26738, 26774, 27674, 27782, 28358, 28718, 28754, 29798, 29942, 31526, 31706, 31922, 32219, 32714, 33002, 33182, 33506, 34046, 34298, 34946, 35486, 36566, 36674, 37178, 37682, 37862, 38222, 38582, 39266, 39842, 40634, 41642, 41822, 42911, 43334, 43406, 43658, 43703, 44594, 45026, 45386, 45782, 45854, 46178, 46646, 47366, 47402, 47618, 48554, 48662, 49346, 49706, 50534, 51319, 51326, 51722, 52217, 52334, 52622, 52838, 53126, 53306, 53486, 53702, 53882, 54098, 54494, 54926, 55178, 55187, 55682, 56078, 56762, 57014, 57662, 58454, 58598, 59102, 59246, 59642, 60038, 60254, 60929, 61622, 61946, 62018, 62198, 62414, 63278, 63638, 63998, 64034, 64322, 64394, 64574, 65186, 65978, 66086, 67454, 67958, 68498, 70586, 70829, 71306, 71522, 72062, 72413, 73106, 73538, 73898, 74762, 75086, 75806, 75878, 76274, 76526, 76627, 76994, 77174, 77858, 78254, 78542, 78578, 78866, 78938, 79046, 79514, 80558, 81422, 81818, 83114, 83897, 84158, 84986, 85382, 85634, 86174, 86246, 86714, 86858, 87326, 87758, 88154, 88334, 89018, 89281, 89441, 89486, 89639, 89738, 90422, 90926, 90998, 92213, 92402, 93122, 93302, 93554, 93698, 94454, 95678, 95786, 96686, 96722, 96758, 97226, 97262, 97442, 98054, 98747, 98846, 99818

However, of these only three have digital roots that are all equal to 1. These are:$$\begin{array}{rl}51319& =19\times 37\times 73\\ 76627& =19\times 37\times 109\\ 89281& =19\times 37\times 127\end{array}$$As can be seen, two of the factors (19 and 37) are the same for all three numbers. If we extend the range to one million, these two factor and 73 (the reversal of 37) make frequent appearances.

If we relax the requirement that the digital roots of the factors must be equal and require only the the digital roots of the factors multiply together to give the digital root of the number, then we find that 1969 numbers satisfy and that will include the 230 numbers mentioned earlier (permalink). An example would be 27813:$$\underset{3}{\underbrace{27813}}=\underset{3}{\underbrace{3}}\times \underset{1}{\underbrace{73}}\times \underset{1}{\underbrace{127}}$$

The Nautical Mile

Today I turned 27780 days old and one of the properties of this number is that it is a multiple of 1852:$$27780=15\times 1852$$The obvious question is so what? Well, 1852 turns out to be the length of a nautical mile in metres and numbers that are multiples of it belong to a special sequence, OEIS A303272. The initial members of the sequence are:

0, 1852, 3704, 5556, 7408, 9260, 11112, 12964, 14816, 16668, 18520, 20372, 22224, 24076, 25928, 27780, 29632, 31484, 33336, 35188, 37040, 38892, 40744, 42596, 44448, 46300, 48152, 50004, 51856, 53708, 55560, 57412, 59264, 61116, 62968, 64820, 66672, 68524

Figure 1

So if one day of my life represents a metre then I have today travelled 15 nautical miles on my journey through life. However, let's look more closely at how this figure of 1852 comes about. See Figure 1 and here are some excerpts from Wikipedia:

By the mid-19th century, France had defined a nautical mile via the original 1791 definition of the metre, one ten-millionth of a quarter meridian. So:$$\frac{10,000,000}{90\times 60}\text{m }=1,851.85\text{ m }\approx 1,852\text{ m }$$became the metric length for a nautical mile. France made it legal for the French Navy in 1906, and many metric countries voted to sanction it for international use at the 1929 International Hydrographic Conference. In 1929 the international nautical mile was defined by the First International Extraordinary Hydrographic Conference in Monaco as exactly 1,852 metres (which is 6,076.12 ft). The United States did not adopt the international nautical mile until 1954. Britain adopted it in 1970. The derived unit of speed is the knot, one nautical mile per hour.

OEIS A303272 suggests a second similar sequence based on 1760, the number of yards in a mile. This would produce the follow initial members:

0, 1760, 3520, 5280, 7040, 8800, 10560, 12320, 14080, 15840, 17600, 19360, 21120, 22880, 24640, 26400, 28160, 29920, 31680, 33440, 35200, 36960, 38720, 40480, 42240, 44000, 45760, 47520, 49280, 51040, 52800, 54560, 56320, 58080, 59840, 61600, 63360, 65120, 66880, 68640, 70400

However, this sequence is not listed in the OEIS. Multiples of 5280, the number of feet in a mile, are included of course in the above sequence of numbers.

Descent to Zero

The number associated with my diurnal age today, 27779, has the interesting property that it is the smallest number that takes 22 steps to reach 0 under "$k\to$ max product of two numbers whose concatenation is $k$". The possible concatenatable pairs and their products for 27779 are:

• 2 * 7779 = 15558
• 27 * 779 = 21033
• 277 * 79 = 21883
• 2777 * 9 = 24993

We see that 24993 is the maximum product and we repeat the process using this number as our starting point:

• 2 * 4993 = 9986
• 24 * 993 = 23832
• 249 * 93 = 23157
• 2499 * 3 = 7497

The maximum product is 23832 and so this becomes the new number and the process continues until we reach 0 after 22 steps. The progression is as follows (permalink):

27779, 24993, 23832, 19136, 11478, 9176, 6916, 5496, 5184, 4284, 3528, 2816, 1686, 1376, 988, 792, 644, 264, 128, 96, 54, 20, 0

27779 is a member of OEIS A035932: smallest number that takes $n$ steps to reach 0 under "$k\to$ max product of two numbers whose concatenation is $k$". The initial members of this sequence are:

0, 1, 11, 26, 39, 77, 117, 139, 449, 529, 777, 1117, 2229, 2982, 4267, 4779, 5319, 5919, 8693, 12699, 14119, 17907, 27779, 47877, 80299, 103199, 135199, 274834, 293938, 312794, 606963, 653993, 773989, 1160892, 1296741, 1616696, 1986576

This permalink will check for the smallest number once the number of steps is specified. When writing the code the convention is that you add a condition to handle single-digit numbers. A common rule for sequences like this that aim to reach 0 is that single-digit numbers (other than 0) map to 0 in the next step. So looking at the above sequence we see following progressions to 0 beginning with 0 that requires zero steps (permalink):

• 0
• 1, 0
• 11, 1, 0
• 26, 12, 2, 0
• 39, 27, 14, 4, 0
• 77, 49, 36, 18, 8, 0
• 117, 77, 49, 36, 18, 8, 0
• 139, 117, 77, 49, 36, 18, 8, 0
• 449, 396, 288, 224, 88, 64, 24, 8, 0
• 529, 468, 368, 288, 224, 88, 64, 24, 8, 0
• 777, 539, 477, 329, 288, 224, 88, 64, 24, 8, 0
• 1117, 777, 539, 477, 329, 288, 224, 88, 64, 24, 8, 0
• 2229, 1998, 1862, 1116, 666, 396, 288, 224, 88, 64, 24, 8, 0
• 2982, 2378, 1896, 1728, 1376, 988, 792, 644, 264, 128, 96, 54, 20, 0

More On Sums of Digits

Not long ago, in December of 2024, I created a post titled Prime Sums Of Digits, Digits Squared And Digits Cubed in which I looked at numbers that have a sum of digits, a sum of digits squared and a sum of digits cubed that are all prime. There are 1985 such numbers in the range up to 40000 and 322 of them are prime themselves.

In this post I want to look at numbers that not only meet the just mentioned criteria but have the additional property that, when these sums are added to the original number, the result is also prime. Here are the criteria that such numbers must meet with $\text{SOD}$ standing for Sum Of Digits:

• $\text{SOD}$ of number is prime
• ${\text{SOD}}^2$ of number is prime
• ${\text{SOD}}^3$ of number is prime
• number + $\text{SOD}$ is prime
• number + ${\text{SOD}}^2$ is prime
• number + ${\text{SOD}}^3$ is prime

As it turns out, there are only 19 such numbers in the range up to 40000 and only the first two of them are prime (permalink): 

11, 101, 166, 4490, 4528, 4630, 6016, 8254, 8788, 10066, 12422, 13166, 18284, 18688, 20854, 25570, 31166, 32518, 36064

The first composite number is 166 so let's check that it satisfies the criteria:

• $\text{SOD}$ is prime: 13
• ${\text{SOD}}^2$ is prime: 73
• ${\text{SOD}}^3$ is prime: 433
• 166 + 13 = 179 is prime
• 166 + 73 = 239 is prime
• 166 + 433 = 599 is prime

Interestingly, six of the above numbers survive the addition of a further criterion, namely that:

•  ${\text{SOD}}^4$ is prime
• number + ${\text{SOD}}^4$ is prime

These numbers are 11, 101, 4528, 6016, 10066, 20854 (permalink) and we have to look at much larger numbers to find any that satisfy yet another criterion, namely:

•  ${\text{SOD}}^5$ is prime
• number + ${\text{SOD}}^5$ is prime

The only two numbers that satisfy this additional criteria are 100001 and 104930. The former number is not prime but composite since 100001 = 11 x 9091. It would be interesting to see how much further this process could be taken but we would then be looking at very large numbers indeed.