Simultaneously Inconsummate, Self and Untouchable Numbers

Today I turn 27444 days old and one property of this number is that it is inconsummate, meaning that there is no number that divided by its sum of digits equals 27444. What I noticed however, was that 27111, 27222, 27333, 27444, 27666, 27777, 27888 and 27999 are all inconsummate. 

Notice that 27555 is not inconsummate because 991980, when divided by its sum of digits (36), gives 27555. Similarly 27000 is not inconsummate because 243000, when divided by its sum of digits (9) gives 27000 as does 486000 when divided by its sum of digits (18) etc. The latter result is to be expected because doubling a number produces the same dividend when it is divided by the sum of digits. Similarly, tripling a number produces the same dividend and so any number that is "consummate" has an infinity of numbers that, when divided by their sum of digits, produce the number.

The pattern is less noticeable in the range from 26000 to 26999 where only 26111, 26666 and 26888 are inconsummate. In the range from 28000 to 28999, none of the 28XXX numbers are inconsummate. In the range from 29000 to 29999, we find 29222, 29555 and 29777 to be inconsummate. So the 27000 to 27999 millenium seems to produce one of the highest counts of ABXXX numbers but whether this is the highest, I don't know. In my post titled Inconsummate Numbers from the 1st of August 2018, I provide a list of all inconsummate numbers from 62 to 65535.

Getting back to the number associated with my diurnal age (27444) we find that it is also:

• a self number, because there is no number that, added to its sum of digits, gives 27444
• an untouchable number, because it is not equal to the sum of proper divisors of any number

One might reasonably ask the question as to how many numbers are inconsummate, self and untouchable? I was able to identify all the numbers up to 40000 with this property and there are 265 of them. Here they are:

872, 2672, 3752, 3818, 3842, 3864, 4046, 4316, 4338, 4382, 4472, 4494, 4742, 4832, 4854, 4898, 5126, 5148, 5372, 6654, 7284, 7598, 8162, 9152, 9218, 9264, 9848, 10076, 10368, 10379, 10412, 10884, 10974, 11481, 11516, 11549, 12594, 12752, 13226, 13259, 13314, 13382, 13742, 13922, 14126, 14148, 14328, 14394, 14418, 14664, 14754, 14798, 14822, 14934, 14978, 15116, 15215, 15452, 15474, 15507, 15597, 16251, 16811, 17217, 17285, 17544, 17588, 17621, 17757, 18141, 18387, 18422, 18837, 20325, 20514, 20874, 20918, 21392, 22316, 22652, 23214, 23664, 23888, 24722, 24755, 25071, 25104, 25317, 25374, 25418, 25464, 25532, 25622, 25655, 25868, 25901, 26039, 26981, 27029, 27420, 27444, 27611, 28142, 28254, 28377, 28388, 28511, 28737, 28748, 28926, 29165, 29187, 29222, 29244, 29321, 29424, 29435, 29760, 29995, 30337, 30348, 30359, 30449, 30651, 30774, 30998, 31057, 31147, 31237, 31292, 31314, 31428, 31439, 31584, 31707, 31764, 31808, 31832, 31922, 31955, 32025, 32036, 32047, 32069, 32091, 32137, 32159, 32214, 32394, 32418, 32429, 32484, 32552, 32574, 32585, 32618, 32732, 32798, 32822, 32855, 32934, 33037, 33059, 33092, 33114, 33171, 33239, 33698, 33911, 34082, 34374, 34385, 34587, 34655, 34699, 34767, 34925, 34947, 35105, 35151, 35318, 35432, 35454, 35577, 35588, 35621, 35757, 35847, 35880, 35891, 35924, 35948, 35981, 36029, 36095, 36207, 36242, 36264, 36275, 36365, 36398, 36422, 36444, 36455, 36488, 36510, 36567, 36578, 36624, 36635, 36701, 36769, 36791, 36813, 36837, 36848, 36859, 36927, 36949, 36960, 36971, 36993, 37015, 37085, 37175, 37197, 37232, 37265, 37298, 37421, 37434, 37478, 37489, 37535, 37568, 37579, 37623, 37680, 37781, 37803, 37827, 37838, 37871, 37926, 37928, 38005, 38154, 38165, 38220, 38310, 38378, 38525, 38760, 38916, 38927, 39144, 39221, 39537, 39548, 39581, 39671, 39704, 39783, 39851, 39917

So 27444 turns out to be rather special and all such numbers are in a sense quite isolated because they cannot be derived by dividing a number by its sum of digits, nor can they be had by adding the sum of a number's digits to the number and finally they cannot be derived from the addition of the proper divisors of any number. See Bespoken for Sequences entry.

Of the 265 numbers above, 20 of them are prime. These are 11549, 13259, 16811, 26981, 27611, 30449, 31147, 31237, 32069, 32159, 32429, 33037, 33911, 36791, 37489, 37579, 37781, 37871, 39581 and 39671. This is about the number you'd expect by chance, even if it is a little on the low side. There are 54 semiprimes. Of numbers with three consecutive digits that are the same, there are 23888, 27444, 29222, 29995 and 36444.

Primes Formed By Concatenation

 Suppose we laid down the following criteria that prime numbers had to adhere to:

• formed from the \(n^{th}\) number and the  \(n^{th}\) prime
• no repeating digits
• sum of digits is a prime number

The criterion that there must be no repeating digits means that we must have a finite number of such primes because, as the primes get larger, digits must repeat. Concatenating the first number 1 and the first prime 2, we get 12 which is not prime. However, concatenating the second number 2 and the second prime 3, we get 23 which satisfies the criteria. The next possibility, 35, doesn't satisfy but 47 does. 

Writing a program that yields all the conforming primes up to ten million, yields the following select group where | represents the operation of concatenation: 

• 2 | 3 --> 23
• 4 | 7 --> 47
• 12 | 37 --> 1237
• 27 | 103 --> 27103
• 57 | 269 --> 57269
• 58 | 271 --> 58271
• 85 | 439 --> 85439
• 93 | 487 --> 93487
• 145 | 829 --> 145829
• 406 | 2791 --> 4062791
• 591 | 4327 --> 5914327
• 835 | 6421 --> 8356421

Interestingly, apart from 2, all the  \(n^{th}\) numbers are composite in the range up to 999. This investigation arose from the number associated with my diurnal age today, namely 27103. It turned out that this number is a member of OEIS A084667:

A084667

Primes which are a concatenation of \(n\) and prime(\(n\)).

For this sequence, we are only applying the first criterion. The initial members of the sequence are shown below with previous primes marked in bold:

23, 47, 613, 1237, 1759, 1861, 2383, 27103, 30113, 35149, 36151, 41179, 42181, 45197, 46199, 54251, 56263, 57269, 58271, 61283, 71353, 82421, 83431, 85439, 92479, 93487, 99523, 115631, 117643, 119653, 121661, 123677, 127709, 136769, 141811, 145829, 147853 

It can be noted that of the terms shown, the  \(n^{th}\) number in several cases is prime e.g. 1759, 2383 etc. I was interested in finding out how many primes survived once the second and third criteria were applied and 27103 survived as can be seen. 

Osculators

In this post I'm describing a very interesting method of determining the divisors of any given number that I came across in a post by Sohel Sahoo:

All credit is given to Sohel Sahoo for his method and all I've done in this post is to rephrase his method so that it is easier for me to understand. 

The assertion is made that there exist two specific numbers (called osculators: one positive, the other negative) for any divisor. The divisor's positive or the negative osculator can be used to determine if a given number has that particular divisor. The sum of the absolute values of the osculators is equal to the divisor.

Procedures to find the osculators of a given divisor

(7 and 13 will be used as examples):

1. If the divisor is a single digit, work with the smallest multiple that has two digits.
   
   In the case of 7, the smallest two digit multiple is 14 so we work with that.
   
   In the case of 13, there is no need to modify it.
   
   
2. Let the osculator be \(x\) and multiply the unit digit of the divisor by \(x\).
   
   In the case of 14, this gives \(4x\).
   
   In the case of 13 this gives \(3x\).
   
   
3. Add the result obtained in step 2 to the remaining digits to obtain an expression in \(x\).
   
   In the case of 14 this gives the expression \(1+4x\).
   
   In the case of 13 this gives the expression \(1+3x\).
   
   
4. Find the smallest positive and negative values of \(x\) that make the expression divisible by the divisor. These are the positive and negative osculators for the divisor.
   
   In the case of 7, \(1+4 \times 5=21\) and \(1+4 \times -2 =-7\) and so the osculators are 5 and -2. Note that |5|+|-2|=7.
   
   In the case of 13, \(1 + 3 \times 4=13\) and \(1+ 3 \times -9 = -26\) and so the osculators are 4 and -9. Note that |4|+|-9|=13.

Doing this for other divisors generates the table shown below:

Number	Positive Osculator	Negative  Osculator
3	1	-2
7	5	-2
9	1	-8
11	10	-1
13	4	-9
17	12	-5
19	2	-17
21	19	-2
23	7	-16
27	19	-8
29	3	-26
31	28	-3
33	10	-23
37	26	-11
39	4	-35
41	37	-4
43	13	-30
47	33	-14
49	5	-44
51	46	-5
53	16	-37
57	40	-17
59	6	-53
61	55	-6
63	19	-44
67	47	-20
69	7	-62
71	64	-7
73	22	-51
77	54	-23
79	8	-71
81	73	-8
83	25	-58
87	61	-26
89	9	-80

Some tips for remembering the osculators:

Any divisor ending in 9 can generally be written as \(a9\).

Let the osculator be \(x\).

According to rules the expression will be \(9x+a\).

In decimal representation we have:

\(a9=10a+9=9a+a+9=9a+9+a=9(a+1)+a\)

Obviously, the positive osculator is (\(a+1\)). Henceforth, owing to this, the positive osculator for divisor numbers ending in 9 is just one more than its previous digits.

1. For 9, 19, 29, 39 etc.(all ending in 9), the positive osculators are 1,2,3,4 etc.
   
   
2. For 3, 13, 23, 33 etc. (all ending in 3) multiply them by 3 in order to get 9 in the unit place as 9, 39, 69, 99 etc. Thus you get 1, 4, 7, 10 etc. as positive osculators.
   
   
3. For 7, 17, 27, 37 etc.(all ending in 7) multiply them by 7 so as to attain 9 in unit place like 49, 119, 189, 259 etc. Hence you get 5, 12, 19, 26 etc. as positive osculators.
   
   
4. For 1, 11, 21, 31 etc. (all ending in 1) multiply them by 9 thereof attain 9 as last digit as 9, 99, 189, 279 etc. So, you get positive osculators viz. 1, 10, 19, 28 etc.

Special Divisibility Rule:

If any number is made by repeating a digit 6 times then the no. will be divisible by 3, 7, 11, 13, 21, 37, 77, 91, 143 and 1001 e.g. 111111, 222222 and 333333 are divisible by these numbers.

Formula for using the osculators to test divisibility:

We'll use 69125 and divisibility by 7 as an example to illustrate the use of the formula. We'll use both the positive and negative osculators (although in practice, only one is needed):

1. Firstly, multiply the osculator of the divisor by the unit digit of the number that is being tested for divisibility by that divisor.
   
   In the case of 69125 being test for divisibility by 7, this means 5 x 5 = 35 using the positive osculator of 5 and 5 x -2 = 10 using the negative osculator of -2.
   
   
2. Add the result so obtained on multiplication to the remaining of digits if using the positive osculator, subtract the result if using the negative osculator.
   
   Thus 69125 --> 6912 + 25 = 6937 or 6912 - 10 = 6902.
   
   
3. Repeat processes 1 and 2 until the number is small enough to be recognised as a multiple of the divisor or not. 
   
   Thus 6937 --> 693 + 35 = 728 --> 72 + 40 = 112 --> 11 + 10 --> 21 which is 7 x 3.
   6902 --> 690 - 4 = 686 --> 68 -12 = 56 which is 7 x 8.
   
   
4. If the result is a multiple of the divisor then the divisor is confirmed.
   
   69125 does reduce to a multiple of 7 and so 7 is a divisor. It can also be seen that using the smaller of the two osculators (ignoring signs) makes for easier calculations.

Another example:

What happens if 69125 were tested for divisibility by 13? We'll use the 4 as the osculator as it is the smaller of the two.

69125 --> 6912 - 20 = 6892 --> 689 - 8 = 681 --> 68 - 4 = 64 which is clearly not a multiple of 13 and so we conclude that 69125 is not divisible by 13.

Usefulness of Divisibility Tests

One can argue that learning divisibility tests like this is pointless in this technological age but I think such mental activities combat mental decline (I am a septuagenarian so that's important) and reduce our reliance on technology by making us more confident and self-sufficient (as a counterbalance to the encroachments of AI).

Note on Osculators and Osculation

It should be noted that the terms osculator and osculation have special meanings in Vedic Mathematics. An osculator is an algorithm for performing osculation while osculation means the determination of whether a number is divisible by another by means of certain operations on its digits. The meaning of osculation is thus different from that of mainstream mathematics where the term means a contact between curves or surfaces, at which point they have a common tangent. Thus we can speak of osculating circles, meaning two circles that touch at a point through which a common tangent passes. Here is a link to more information on Vedic Mathematics.