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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