# Friendly numbers

Two different natural numbers , one of which is mutually equal to the sum of the real divisors of the other number, form a pair of friendly numbers .

The sum of the real factors of is often referred to as . The definition can thus also be formulated as follows: ${\ displaystyle x}$${\ displaystyle \ sigma ^ {*} (x)}$

Two different natural numbers and form a pair of friendly numbers if: and . ${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle \ sigma ^ {*} (a) = b}$${\ displaystyle \ sigma ^ {*} (b) = a}$

## Examples

• The smallest friendly pair of numbers is formed by the numbers 220 and 284. It is easy to calculate that the two numbers meet the definition:
• The sum of the real divisors of 220 is ${\ displaystyle 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284}$
• and gives the sum of the real divisors of 284 .${\ displaystyle 1 + 2 + 4 + 71 + 142 = 220}$
• The first friendly pairs of numbers are the following:
• (220, 284), (1184, 1210), (2620, 2924), (5020, 5564), (6232, 6368), (10744, 10856), (12285, 14595), (17296, 18416), (63020 , 76084), (66928, 66992), (67095, 71145), (69615, 87633), (79750, 88730), ... (sequence A259180 in OEIS ) or (sequence A002025 in OEIS ) and (sequence A002046 in OEIS )
• Most known friendly number pairs have a common smallest prime factor (usually a 2 or a 5). But there are friendly pairs of numbers that have no common smallest prime factor. Seven such pairs are currently known (as of January 30, 2016), the smallest pair is the following:
• ${\ displaystyle 445953248528881275 = 3 ^ {2} \ times 5 ^ {2} \ times 7 \ times 13 \ times 19 \ times 37 \ times 43 \ times 73 \ times 439 \ times 22483}$
• ${\ displaystyle 659008669204392325 = 5 ^ {2} \ times 7 \ times 13 \ times 19 \ times 37 \ times 73 \ times 571 \ times 1693 \ times 5839}$
• Many friendly pairs of numbers have a sum that is divisible by 10. The first of these pairs of numbers are the following:
• (6232, 6368), (10744, 10856), (12285, 14595), (66928, 66992), (67095, 71145), (79750, 88730), (100485, 124155), (122265, 139815), (122368 , 123152), (141664, 153176), (142310, 168730), (176272, 180848), (185368, 203432), (356408, 399592), (437456, 455344),… (sequence A291422 in OEIS )

## Properties and unsolved problems

• In a friendly pair of numbers, the smaller number is always abundant and the larger number deficient .
• The density of friendly numbers is 0.
• So far, friendly numbers are either both odd or both even. It is not yet known if there are friendly numbers where one number is odd and the other number is even. If such a pair of numbers exists, the even number must either be a square number or twice a square number. The odd number must be a square number.
• Every known pair of friendly numbers has at least one prime factor in common. It is not yet known whether there are friendly pairs of numbers that are relatively prime . If such a pair of numbers exists, the product of the two numbers must be at least . Such a pair of numbers can neither be generated by Thabit's formula (in the text below) nor by a similar formula.${\ displaystyle (a, b)}$${\ displaystyle a \ cdot b> 10 ^ {67}}$

## Early mentions and the Thabit Ibn Qurra sentence

Pythagoras was first mentioned around 500 BC. The friendly numbers 220 and 284. When asked what a friend was, he replied: "One who is a different me, like 220 and 284."

In 1636, Pierre de Fermat informed Marin Mersenne in a letter that he had found the friendly numbers 17296 and 18416. However, Walter Borho determined in 2003 that this pair of numbers was found by Ibn al-Banna (1265–1321) and Kamaladdin Farist in the 14th century . Ibn al-Banna is quoted as saying: "The numbers 17296 and 18416 are friendly, one abundant , the other deficient . Allah is omniscient."

The sentence of Thabit Ibn Qurra was used :

For a fixed natural number let ${\ displaystyle n}$
${\ displaystyle x = 3 \ cdot 2 ^ {n} -1}$
${\ displaystyle y = 3 \ cdot 2 ^ {n-1} -1}$
${\ displaystyle z = 9 \ cdot 2 ^ {2n-1} -1}$.
If and are prime numbers, then the two numbers and are friends.${\ displaystyle x, y}$${\ displaystyle z}$${\ displaystyle a = 2 ^ {n} \ cdot x \ cdot y}$${\ displaystyle b = 2 ^ {n} \ cdot z}$

The proof of this theorem can be found in the article on divisional sums .

Numbers of the form are therefore also called Thabit numbers . Two consecutive Thabit numbers must be prime, which limits the possible values ​​for very. ${\ displaystyle 3 \ cdot 2 ^ {n} -1}$${\ displaystyle n}$

### Examples

• For are all prime numbers. This results in${\ displaystyle n = 2}$${\ displaystyle x = 11, y = 5, z = 71}$
${\ displaystyle a = 4 \ times 11 \ times 5 = 220}$
${\ displaystyle b = 4 \ cdot 71 = 284}$
• For is not prime, i.e. H. with you don't find friendly numbers.${\ displaystyle n = 3}$${\ displaystyle z = 287 = 7 \ cdot 41}$${\ displaystyle n = 3}$
• The couple that Fermat has found are friends .${\ displaystyle n = 4}$${\ displaystyle (17296,18416)}$
• For Descartes calculated in 1638 the befriended couple . However, according to Borho, these had also been determined beforehand by Muhammad Baqir Yazdi .${\ displaystyle n = 7}$ ${\ displaystyle (9,363,584,9,437,056)}$

Today it is known that one cannot determine any other friendly numbers for with Thabit's theorem . ${\ displaystyle n \ leq 191,600}$

## A sentence by Leonhard Euler

Leonhard Euler generalized Thabit's theorem:

For a fixed natural number let ${\ displaystyle n \ in \ mathbb {N}}$
${\ displaystyle x = f \ cdot 2 ^ {n} -1}$
${\ displaystyle y = f \ cdot 2 ^ {nk} -1}$
${\ displaystyle z = f ^ {2} \ cdot 2 ^ {2n-k} -1}$
with and .${\ displaystyle f = 2 ^ {k} +1}$${\ displaystyle n> k> 0}$
If and are prime numbers, then the two numbers and are friends.${\ displaystyle x, y}$${\ displaystyle z}$${\ displaystyle a = 2 ^ {n} \ cdot x \ cdot y}$${\ displaystyle b = 2 ^ {n} \ cdot z}$

For the special case one obtains the Thabit theorem. ${\ displaystyle k = 1}$

In 1747 Euler found 30 more friendly pairs of numbers and published them in his work De numeris amicabilibus . 3 years later he published a further 34 pairs of numbers, but 2 of them were wrong.

In 1830 Adrien-Marie Legendre found another couple.

In 1866 the Italian B. Niccolò I. Paganini (not the violin virtuoso ) showed as a 16-year-old that 1184 and 1210 are friendly numbers. Until then, this had been overlooked. It is the second smallest friendly pair of numbers.

In 1946, Escott published the complete list of the 390 friendly pairs of numbers that were known until 1943.

In 1985, Herman te Riele (Amsterdam) calculated all friendly numbers less than 10,000,000,000 - a total of 1427 couples.

In 2007 almost 12 million pairs of numbers were known to be friends.

In May 2018, 1,222,206,716 friendly pairs of numbers were known.

It is believed that there are an infinite number of friendly numbers, but no evidence is known yet.

## Walter Borho's Theorem

More friendly numbers can be found with the help of Walter Borho's theorem :

Be and be friends with numbers with and , where is a prime number.${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle A = a \ cdot u}$${\ displaystyle B = a \ cdot s}$${\ displaystyle s \ in \ mathbb {P}}$
Continue to be a prime number and not a divisor of .${\ displaystyle p = u + s + 1}$${\ displaystyle p}$${\ displaystyle a}$
Then:
If the two numbers are prime and prime for a fixed natural number , then and are friendly numbers.${\ displaystyle n}$${\ displaystyle q_ {1} = (u + 1) \ cdot p ^ {n} -1}$${\ displaystyle q_ {2} = (u + 1) \ cdot (s + 1) \ cdot p ^ {n} -1}$${\ displaystyle A_ {1} = Ap ^ {n} q_ {1}}$${\ displaystyle B_ {1} = ap ^ {n} q_ {2}}$

Example:

${\ displaystyle A = 220 = 2 ^ {2} \ cdot 55}$and are friends. So are and , where is prime.${\ displaystyle B = 284 = 2 ^ {2} \ cdot 71}$${\ displaystyle a = 4, u = 55}$${\ displaystyle s = 71}$${\ displaystyle s}$
${\ displaystyle p = 127}$is prime and not a divisor of . ${\ displaystyle a = 4}$
• ${\ displaystyle {\ underline {n = 1}}}$:
${\ displaystyle q_ {1} = 56 \ cdot 127-1 = 7111 = 13 \ cdot 547}$is not prime. For therefore we obtain no new amicable numbers.${\ displaystyle n = 1}$
• ${\ displaystyle {\ underline {n = 2}}}$:
${\ displaystyle q_ {1} = 56 \ cdot 127 ^ {2} -1 = 903,223}$and are both prime. It follows:${\ displaystyle q_ {2} = 56 \ cdot 72 \ cdot 127 ^ {2} -1 = 65,032,127}$
${\ displaystyle A_ {1} = 220 \ times 127 ^ {2} \ times 903.223 = 4.195.612.705.532}$and are friendly numbers.${\ displaystyle B_ {1} = 4 \ cdot 127 ^ {2} \ cdot 65,032,127 = 3,204,978,428,740}$

With the help of this theorem, Borho found another 10,455 friendly numbers.

## Regular pairs of friendly numbers

Be a friendly pair of numbers with and be and with (so it is the greatest common factor of and ). If both and coprime to and are square-free , then one calls a regular friendly number pair . ${\ displaystyle (A, B)}$${\ displaystyle A ${\ displaystyle A = g \ cdot a}$${\ displaystyle B = g \ cdot b}$${\ displaystyle \ operatorname {ggT} (A, B) = g}$${\ displaystyle g}$${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle a}$${\ displaystyle b}$ ${\ displaystyle g}$${\ displaystyle (A, B)}$

The first number of the smallest regular friendly number pairs are:

220, 2620, 5020, 10744, 17296, 63020, 66928, 67095, 69615, 100485, 122265, 142310, 171856, 176272, 185368, 196724, 308620, 356408, 437456, 503056, 522405, 600392, 609928, 624184, 635624, 643336, 667964, 726104, 898216, 947835, 998104, 1077890, ... (sequence A215491 in OEIS )

If a friendly pair of numbers is not regular, then it is an irregular, friendly pair of numbers (or an exotic, friendly pair of numbers ).

If, in a regular friendly number pair, the first number has exactly prime factors and the second number has exactly prime factors, then the regular friendly number pair is of the type . ${\ displaystyle (A, B)}$${\ displaystyle A}$${\ displaystyle i}$${\ displaystyle B}$${\ displaystyle j}$${\ displaystyle (i, j)}$

Examples:

• For the two figures from the friendly pair of numbers applies: . So is and . So is and . It has exactly two prime factors and exactly one prime factor. Thus the friendly pair of numbers is of regular type .${\ displaystyle (A, B) = (220.284)}$${\ displaystyle g = \ operatorname {ggT} (A, B) = 4}$${\ displaystyle A = g \ cdot a = 4 \ cdot 55}$${\ displaystyle B = g \ cdot b = 4 \ cdot 71}$${\ displaystyle a = 55 = 5 \ cdot 11}$${\ displaystyle b = 71 \ in \ mathbb {P}}$${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle (A, B) = (220.284)}$${\ displaystyle (2,1)}$
• For the two figures from the friendly pair of numbers applies: . So is and . But it is neither coprime nor square-free, and it is not square-free either. Thus, the friendly pair of numbers is an irregular friendly pair of numbers.${\ displaystyle (A, B) = (1184.1210)}$${\ displaystyle g = \ operatorname {ggT} (A, B) = 2}$${\ displaystyle A = g \ times a = 2 \ times 592}$${\ displaystyle B = g \ cdot b = 2 \ cdot 605}$${\ displaystyle a = 592 = 2 ^ {4} \ cdot 37}$${\ displaystyle g = 2}$${\ displaystyle b = 605 = 5 \ cdot 11 ^ {2}}$${\ displaystyle (A, B) = (1184.1210)}$

## Pair of twins friends

A friendly pair of numbers is befriended twins ( twin amicable pairs ) if there is no integers between and are belonging to another friendly pair of numbers. ${\ displaystyle (a, b)}$${\ displaystyle a}$${\ displaystyle b}$

The first befriended twin pairs are the following:

(220, 284), (1184, 1210), (2620, 2924), (5020, 5564), (6232, 6368), (10744, 10856), (12285, 14595), (17296, 18416), (66928 , 66992), (122368, 123152), (196724, 202444), (437456, 455344), (469028, 486178), (503056, 514736), (522405, 525915), (643336, 652664), (802725, 863835 ), (998104, 1043096), (1077890, 1099390),… (Follow A273259 in OEIS )

Example:

The friendly pair of numbers is not a friendly pair of twins because, for example, the first number of the friendly pair of numbers is between and . Thus the friendly pair of numbers is not a friendly pair of twins either. The friendly pair of numbers is even completely between and , but is still a friendly pair of twins because there is no other number between and that belongs to a friendly pair of numbers. ${\ displaystyle (63020,76084)}$${\ displaystyle (69615,87633)}$${\ displaystyle 69615}$${\ displaystyle 63020}$${\ displaystyle 76084}$${\ displaystyle (69615,87633)}$ ${\ displaystyle (66928,66992)}$${\ displaystyle 63020}$${\ displaystyle 76084}$${\ displaystyle 66928}$${\ displaystyle 66992}$

## generalization

As already mentioned at the beginning of this article, friendly pairs of numbers have the property that and . If you take all the factors (not only the real ones, but also the number itself), then the following applies . This property can be generalized: ${\ displaystyle (a, b)}$${\ displaystyle \ sigma ^ {*} (a) = b}$${\ displaystyle \ sigma ^ {*} (b) = a}$${\ displaystyle \ sigma (a) = \ sigma (b) = a + b}$

Let be a number tuple with the following property: ${\ displaystyle (n_ {1}, n_ {2}, \ ldots, n_ {k})}$

${\ displaystyle \ sigma (n_ {1}) = \ sigma (n_ {2}) = \ ldots = \ sigma (n_ {k}) = n_ {1} + n_ {2} + \ ldots + n_ {k} }$

Then the tuple is called a friendly number tuple . ${\ displaystyle (n_ {1}, n_ {2}, \ ldots, n_ {k})}$

Example:

The number tuple is a friendly number triple because the sum of all divisors (including the number itself) for all three numbers always results in the number . ${\ displaystyle (1980,2016,2556)}$${\ displaystyle 6552 = 1980 + 2016 + 2556}$

The number quadruple is a friendly number quadruple because the sum of all divisors (including the number itself) for all four numbers always results in the number . ${\ displaystyle (3270960,3361680,3461040,3834000)}$${\ displaystyle 13927680 = 3270960 + 3361680 + 3461040 + 3834000}$

## Related number classes

### Quasi-friendly numbers

In addition to the friendly numbers, there is another class of numbers that is similar to the friendly numbers: the quasi-friendly numbers. They differ from the friendly numbers in that their divisors do not take into account the number itself as well as the 1, i.e. only the nontrivial divisors .

Example:

48 has the divisors 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48. The number 75 has the factors 1, 3, 5, 15, 25 and 75. The sum of the nontrivial divisors of 48 is , and is the sum of the nontrivial divisors of 75 . ${\ displaystyle 2 + 3 + 4 + 6 + 8 + 12 + 16 + 24 = 75}$${\ displaystyle 3 + 5 + 15 + 25 = 48}$

The first quasi-friendly pairs of numbers are:

(48, 75), (140, 195), (1050, 1925), (1575, 1648), (2024, 2295), (5775, 6128), (8892, 16587), (9504, 20735), (62744 , 75495), (186615, 206504), (196664, 219975), (199760, 309135), (266000, 507759), (312620, 549219), (526575, 544784), (573560, 817479), (587460, 1057595 ), (1000824, 1902215), (1081184, 1331967),… (sequence A005276 in OEIS )

### Sociable numbers

Is a chain (finite sequence) of more than two integers before, each of which the sum of the proper divisors of the predecessor and the first number is the sum of the proper divisors of the last number, it is called (Engl. Of social figures sociable numbers ) . There are currently (as of November 2017) chains of the order (length) 4, 5, 6, 8, 9 and 28 known.

• Example of a chain of order 4 (in November 2017 5398 chains were known):
1,264,460, 1,547,860, 1,727,636, 1,305,184
• Example of a chain of order 5 (the only known one at the moment):
12,496, 14,288, 15,472, 14,536, 14,264
• Example of a chain of order 6 (currently five are known):
21,548,919,483, 23,625,285,957, 24,825,443,643, 26,762,383,557, 25,958,284,443, 23,816,997,477
• Example of a chain of order 8 (currently four are known):
1,095,447,416, 1,259,477,224, 1,156,962,296, 1,330,251,784, 1,221,976,136, 1,127,671,864, 1,245,926,216, 1,213,138,984
• Example of a chain of order 9 (the only known chain at the moment):
805,984,760, 1,268,997,640, 1,803,863,720, 2,308,845,400, 3,059,220,620, 3,367,978,564, 2,525,983,930, 2,301,481,286, 1,611,969,514
• Example of a chain of order 28 (the only known one at the moment):
14,316, 19,116, 31,704, 47,616, 83,328, 177,792, 295,488, 629,072, 589,786, 294,896, 358,336, 418,904, 366,556, 274,924, 275,444, 243,760, 376,736, 381,028, 97,976, 52,946, 45,946, 97,976, 22,946, 122,410.9, 97,976, 152,990, 122,410.976.976 22,744, 19,916, 17,716

In November 2017, a total of 5410 of these chains were known.

Under aliquot sequences (contents chains) shall mean those sequences in which the sum of the proper divisor of a follower member is equal to the succeeding member. The sociable numbers thus form periodic aliquot sequences.