System’s question

What about

1872870801843041394471000000000

Nothing special, this is the number that is in the Simi numeral system

99999

 

Siminumeral system

 

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Aristarchus sequences

While writing my book The Numbers, I came up with dozens of more or less interesting sequences. I wrote some for myself, not others, but in each case I had the intention of returning to them “almost once”.

So in the days I took out one and discovered some very interesting qualities in him and - to my delight - all the more mystery.

Since it’s a whole family of sequences, I felt it would be nice to give them a name. So I started calling myself the Aristarchus sequences. He who accepts this designation accepts who does not. In any case, Aristarchus deserves our respect.

But let's move on!

The basic sequences is formed according to the following rule:

a(1) = 1, if n> 1, a(n) = a(n-1)/gcd(a (n-1), n) if gcd(a(n-1), n) > 1, if gcd(a(n-1), n) = 1, then a(n) = a(n-1) + n.

In words, if the largest common divisor of the previous member and n is greater than 1) so is a real divisor), then we divide the previous member by that, and this becomes the new member, but if the previous member and n are relative prime, the new member is the sum of the previous member and n.

There are no special “screws” in the rule itself, we can wait with interest for the result.

Well, that might even seem disappointing, because this sequences looks like this:

1, 3, 1, 5, 1, 7, 1, 9, 1, ….

That is, it is fairly unanimous and predictable.

If that were all, we might even forget the sequences.

But let’s see what happens if the first member is not 1, but - for the sake of order - 2!

A little surprise. The sequences now looks like this:

2, 1, 4, 1, 6, 1, 8, 1, …

Exemplary order and discipline, almost beautiful! While every sequences would be like this, we would have no problem with them.

But the mathematician is curious, let’s see how this sequences looks like for other novice members when we talk about it so much!

Let's see what the sequences is like if the first member is 3!

Well, you should hold on now! The sequences looks like this:

3, 5, 8, 2, 7, 13, 20, 5, 14, 7, 18, 3, 16, 8, 23, 39, 56, 28, 47, 67, 88, 4, 27, 9, 34, 17, 44, 11, 40, 4, 35, 67, 100, 50, 10, 5, 42, 21, 7, 47, 88, 44, 87, 131, 176, 88, 135, 45, 94, 47, 98, 49, 102, 17, 72, 9, 3, 61, 120, 2, 63, 125, 188, 47, 112, 56, 123, 191, 260, 26, 97, 169, 242, 121, 196, 49, 7, 85, 164, 41, 122, 61, 144, 12, 97, 183, 61, 149, 238, 119, 17, 109, 202, 101, 196, 49, 146, 73, 172, 43, 144, 24, 127, 231, 11, 117, 224, 56, 165, 3, …

Undoubtedly, this looks staggering after the first two cases. You could say a real random sequence. Indeed, what connections can be discovered here? (Warning: this is just a provocative question.)

And now may come the real surprise! Right here, where I left off, this bizarre unruly sequence returns to the well-known chiseled bed and continues like this:

1, 113, 1, 115, 1, 117, 1, 119, 1, …

The case is, to put it mildly, interesting, and once discovered, an honest numerator does not leave it at that, but examines it with all natural numbers. (Not big so, this is the smallest infinite set.)

What a denial, this is also an interesting sequence when the sequences returns to normal.

Here are the modest first results (here I will only give the results that form a continuous line):

1, 2, 111, 7, 5, 3, 25, 22, 25, 111, 111, 4, 7, 5, 5, 6, 22, …

Why does 111 occur so many times, why does it occur twice in a row (if the sequences starts with 10 or 11)?

Dozens of questions, dozens of mysteries. We will certainly never get an answer to some of them, yes to others. And I'm looking forward to them.

Finally, another important additive to liven up life in the world of the Aristarchus sequences. Well, a small change in the basic formula dramatically changes its character, leaving no trace of this old monotonous bed. And this change is that if gcd is 1, then the new term is the sum of the previous and n, minus 1. Then the sequence looks like this:

1, 2, 4, 1, 5, 10, 16, 2, 10, 1, 11, 22, 34, 17, 31, 46, 62, 31, 49, 68, 88, 4, 26, 13, 37, 62, 88, 22, 50, 5, 35, 66, 2, 1, 35, 70, 106, 53, 91, 130, 170, 85, 127, 170, 34, 17, 63, 21, 3, 52, 102, 51, 103, 156, 210, 15, 5, 62, 120, 2, 62, 1, 63, 126, 190, 95, 161, 228, 76, 38, 108, 3, 75, 148, 222, 111, 187, 264, 342, 171, 19, …

We see here that the appearance of 1 does not lead the sequences anywhere, boredom is ruled out.

That is now the first news of the Aristarchus sequences.

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Self-powered numbers

We recently asked the not-so-complicated question: why is the

387 420 489

number interesting?

The answer is: because it is the largest number that can be written with two digits:

387 420 489 = 9⁹

There are people who, if they are not in a good mood, do not see anything interesting in anything. They can be heartily regretted. Let’s keep up our good humor and try now to see something more interesting in this issue!

Well, I’m not sparking interest: the amazing thing is that 9⁹ is properly written out of nine digits. Could this be typical of nⁿ type numbers? What a pleasant start: 1¹ is a single digit. Capture us with excitement…

But the series doesn’t go on like that, it’s stuttering locally. 2² also consists of one digit. This disadvantage can no longer be worked out – until 8. 8⁸ consists of 8 digits.

For 10¹⁰, 10 zeros is enough, but it still need 11 digits. From now on, however, there is no stopping. Above this, to describe nⁿ, it would need an increasing number of digits that n. However, this sequence does not give a dizzying progression. Monotonously growing but gently growing. We do not give examples, but show the first 80 values. Interesting sequence. Unless you're in the mood.

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Friendly neighbors

There are no two more different numbers than two adjacent numbers (i.e., than n and n+1, to make us feel: we are in the field of mathematics). If one number is divisible by a third, it is certain that its neighbor is not. And vice versa. Perfect antipodes.

Is that wrong? What do the properties of numbers mean in the end? Earth is n kilometers from the Sun, not long after that distance is n + 1. Everyone may ask: and then what has changed?

Well, that adds to the greatness of mathematics in that it does not deal with such unnecessary questions, but unbrokenly examines everything that can be found in the field of the properties of numbers.

Therefore, we are also now looking at the question: is there still nothing that connects at least some neighboring numbers, such as a common hobby connecting two neighbors in the city?

Modest merit, but still something if the sum of two neighboring numbers is prime. After all, this is not the case for all consecutive numbers. For example, the sum of 5 and 6 is prime (11), but 7 and 8 is not (15). We can call such numbers P-friendly neighbors. Nothing is easier to define:: all primes (and by implication only those) clearly define such a P-friendly pair of numbers. Otherwise, the property may be somewhat reminiscent of the Goldbach conjecture (one states of even numbers and the other of prime numbers that it can be written as the sum of certain two numbers). The difference is that the first is a presumably unprovable conjecture, but in innumerable true cases it can be solved in several ways, and the second is an easily provable theorem, which in turn has a single solution in specific cases.

A somewhat similar property is when the sum of two adjacent numbers is a square number. Then we can say that the two numbers are Q-friendly numbers. Here it is easy to notice that the sum of two adjacent numbers can never give an even square number, and we can boldly generalize this serious observation: any sum of two adjacent numbers is odd, which can be reversed: every odd number can be written - in a clear way - as the sum of two adjacent numbers.

Starting from the sum of the neighbors, we can endlessly define different similar properties, but now let’s look at a more exciting option. As mentioned, all two neighboring numbers are totally different with respect to their divisors. But is it possible that their total number of divisors is the same (they could be D-friends). The answer is yes, and it is not hard to get stuck quickly first, including the first non-trivial examples. After all, in such a “trivial” way, 2 and 3 are already D-friends.

Returning here to the even-odd question: it is easy to notice that the P- and D-friendly numbers are even and odd, and presumably no substantial regularity can be discovered in this respect. In contrast, Q-friendly numbers are all divisible by 4 (but not always by 8).

We will continue the topic. We would also be happy to share the results of others in this regard.

Finally, we give the first hundred numbers corresponding to the above three properties

P-friendly numbers: 1, 2, 3, 5, 6, 8, 9, 11, 14, 15, 18, 20, 21, 23, 26, 29, 30, 33, 35, 36, 39, 41, 44, 48, 50, 51, 53, 54, 56, 63, 65, 68, 69, 74, 75, 78, 81, 83, 86, 89, 90, 95, 96, 98, 99, 105, 111, 113, 114, 116, 119, 120, 125, 128, 131, 134, 135, 138, 140, 141, 146, 153, 155, 156, 158, 165, 168, 173, 174, 176, 179, 183, 186, 189, 191, 194, 198, 200, 204, 209, 210, 215, 216, 219, 221, 224, 228, 230, 231, 233, 239, 243, 245, 249, 251, 254, 260, 261, 270, 273

Q-friendly numbers: 4, 12, 24, 40, 60, 84, 112, 144, 180, 220, 264, 312, 364, 420, 480, 544, 612, 684, 760, 840, 924, 1012, 1104, 1200, 1300, 1404, 1512, 1624, 1740, 1860, 1984, 2112, 2244, 2380, 2520, 2664, 2812, 2964, 3120, 3280, 3444, 3612, 3784, 3960, 4140, 4324, 4512, 4704, 4900, 5100, 5304, 5512, 5724, 5940, 6160, 6384, 6612, 6844, 7080, 7320, 7564, 7812, 8064, 8320, 8580, 8844, 9112, 9384, 9660, 9940, 10224, 10512, 10804, 11100, 11400, 11704, 12012, 12324, 12640, 12960, 13284, 13612, 13944, 14280, 14620, 14964, 15312, 15664, 16020, 16380, 16744, 17112, 17484, 17860, 18240, 18624, 19012, 19404, 19800, 20200

D-friendly numbers: 1, 2, 14, 21, 26, 33, 34, 38, 44, 57, 75, 85, 86, 93, 94, 98, 104, 116, 118, 122, 133, 135, 141, 142, 145, 147, 158, 171, 177, 189, 201, 202, 205, 213, 214, 217, 218, 230, 231, 242, 243, 244, 253, 285, 296, 298, 301, 302, 326, 332, 334, 344, 374, 375, 381, 387, 393, 394, 429, 434, 445, 446, 453, 481, 501, 507, 514, 526, 537, 542, 548, 553, 565, 603, 604, 609, 622, 633, 634, 645, 663, 664, 694, 697, 698, 706, 717, 724, 735, 741, 745, 766, 776, 778, 782, 793, 802, 805, 817, 819

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Question – This number wit fewer difitd?

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