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Friendly number

In number theory, friendly numbers are two or more natural numbers with a common abundancy index, the ratio between the sum of divisors of a number and the number itself. Two numbers with the same "abundancy" form a friendly pair; n numbers with the same "abundancy" form a friendly n-tuple.

Being mutually friendly is an equivalence relation, and thus induces a partition of the positive naturals into clubs (equivalence classes) of mutually "friendly numbers".

A number that is not part of any friendly pair is called solitary.

The "abundancy" index of n is the rational number σ(n) / n, in which σ denotes the sum of divisors function. A number n is a "friendly number" if there exists mn such that σ(m) / m = σ(n) / n. "Abundancy" is not the same as abundance, which is defined as σ(n) − 2n.

"Abundancy" may also be expressed as ${\displaystyle \sigma _{-\!1}(n)}$ where ${\displaystyle \sigma _{k}}$ denotes a divisor function with ${\displaystyle \sigma _{k}(n)}$ equal to the sum of the k-th powers of the divisors of n.

The numbers 1 through 5 are all solitary. The smallest "friendly number" is 6, forming for example, the "friendly" pair 6 and 28 with "abundancy" σ(6) / 6 = (1+2+3+6) / 6 = 2, the same as σ(28) / 28 = (1+2+4+7+14+28) / 28 = 2. The shared value 2 is an integer in this case but not in many other cases. Numbers with "abundancy" 2 are also known as perfect numbers. There are several unsolved problems related to the "friendly numbers".

In spite of the similarity in name, there is no specific relationship between the friendly numbers and the amicable numbers or the sociable numbers, although the definitions of the latter two also involve the divisor function.

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Examples

As another example, 30 and 140 form a friendly pair, because 30 and 140 have the same "abundancy":

${\displaystyle {\tfrac {\sigma (30)}{30}}={\tfrac {1+2+3+5+6+10+15+30}{30}}={\tfrac {72}{30}}={\tfrac {12}{5}}}$
${\displaystyle {\tfrac {\sigma (140)}{140}}={\tfrac {1+2+4+5+7+10+14+20+28+35+70+140}{140}}={\tfrac {336}{140}}={\tfrac {12}{5}}.}$

The numbers 2480, 6200 and 40640 are also members of this club, as they each have an "abundancy" equal to 12/5.

For an example of odd numbers being friendly, consider 135 and 819 ("abundancy" 16/9). There are also cases of even being "friendly" to odd, such as 42 and 544635 ("abundancy" 16/7). The odd "friend" may be less than the even one, as in 84729645 and 155315394 ("abundancy" 896/351).

A square number can be friendly, for instance both 693479556 (the square of 26334) and 8640 have "abundancy" 127/36 (this example is accredited to Dean Hickerson).

Status for small n

Blue numbers are proved friendly (sequence A074902 in the OEIS), darkred numbers are proved solitary (sequence A095739 in the OEIS), numbers n such that n and ${\displaystyle \sigma (n)}$ are coprime (sequence A014567 in the OEIS) are not coloured darkred here, though they are known to be solitary. Other numbers have unknown status and are highlighted yellow.

 n ${\displaystyle \sigma (n)}$ ${\displaystyle {\frac {\sigma (n)}{n}}}$ n ${\displaystyle \sigma (n)}$ ${\displaystyle {\frac {\sigma (n)}{n}}}$ n ${\displaystyle \sigma (n)}$ ${\displaystyle {\frac {\sigma (n)}{n}}}$ n ${\displaystyle \sigma (n)}$ ${\displaystyle {\frac {\sigma (n)}{n}}}$ 1 1 1 37 38 38/37 73 74 74/73 109 110 110/109 2 3 3/2 38 60 30/19 74 114 57/37 110 216 108/55 3 4 4/3 39 56 56/39 75 124 124/75 111 152 152/111 4 7 7/4 40 90 9/4 76 140 35/19 112 248 31/14 5 6 6/5 41 42 42/41 77 96 96/77 113 114 114/113 6 12 2 42 96 16/7 78 168 28/13 114 240 40/19 7 8 8/7 43 44 44/43 79 80 80/79 115 144 144/115 8 15 15/8 44 84 21/11 80 186 93/40 116 210 105/58 9 13 13/9 45 78 26/15 81 121 121/81 117 182 14/9 10 18 9/5 46 72 36/23 82 126 63/41 118 180 90/59 11 12 12/11 47 48 48/47 83 84 84/83 119 144 144/119 12 28 7/3 48 124 31/12 84 224 8/3 120 360 3 13 14 14/13 49 57 57/49 85 108 108/85 121 133 133/121 14 24 12/7 50 93 93/50 86 132 66/43 122 186 93/61 15 24 8/5 51 72 24/17 87 120 40/29 123 168 56/41 16 31 31/16 52 98 49/26 88 180 45/22 124 224 56/31 17 18 18/17 53 54 54/53 89 90 90/89 125 156 156/125 18 39 13/6 54 120 20/9 90 234 13/5 126 312 52/21 19 20 20/19 55 72 72/55 91 112 16/13 127 128 128/127 20 42 21/10 56 120 15/7 92 168 42/23 128 255 255/128 21 32 32/21 57 80 80/57 93 128 128/93 129 176 176/129 22 36 18/11 58 90 45/29 94 144 72/47 130 252 126/65 23 24 24/23 59 60 60/59 95 120 24/19 131 132 132/131 24 60 5/2 60 168 14/5 96 252 21/8 132 336 28/11 25 31 31/25 61 62 62/61 97 98 98/97 133 160 160/133 26 42 21/13 62 96 48/31 98 171 171/98 134 204 102/67 27 40 40/27 63 104 104/63 99 156 52/33 135 240 16/9 28 56 2 64 127 127/64 100 217 217/100 136 270 135/68 29 30 30/29 65 84 84/65 101 102 102/101 137 138 138/137 30 72 12/5 66 144 24/11 102 216 36/17 138 288 48/23 31 32 32/31 67 68 68/67 103 104 104/103 139 140 140/139 32 63 63/32 68 126 63/34 104 210 105/52 140 336 12/5 33 48 16/11 69 96 32/23 105 192 64/35 141 192 64/47 34 54 27/17 70 144 72/35 106 162 81/53 142 216 108/71 35 48 48/35 71 72 72/71 107 108 108/107 143 168 168/143 36 91 91/36 72 195 65/24 108 280 70/27 144 403 403/144

Solitary numbers

A number that belongs to a singleton club, because no other number is "friendly" with it, is a solitary number. All prime numbers are known to be solitary, as are powers of prime numbers. More generally, if the numbers n and σ(n) are coprime – meaning that the greatest common divisor of these numbers is 1, so that σ(n)/n is an irreducible fraction – then the number n is solitary (sequence A014567 in the OEIS). For a prime number p we have σ(p) = p + 1, which is co-prime with p.

No general method is known for determining whether a number is "friendly" or solitary. The smallest number whose classification is unknown is 10; it is conjectured to be solitary. If it is not, its smallest friend is at least ${\displaystyle 10^{30}}$.[1][2] Small numbers with a relatively large smallest friend do exist: for instance, 24 is "friendly", with its smallest friend 91,963,648.[1][2]

Large clubs

It is an open problem whether there are infinitely large clubs of mutually "friendly" numbers. The perfect numbers form a club, and it is conjectured that there are infinitely many perfect numbers (at least as many as there are Mersenne primes), but no proof is known. As of December 2018, 51 perfect numbers are known, the largest of which has more than 49 million digits in decimal notation. There are clubs with more known members: in particular, those formed by multiply perfect numbers, which are numbers whose "abundancy" is an integer. As of early 2013, the club of "friendly" numbers with "abundancy" equal to 9 has 2094 known members.[3] Although some are known to be quite large, clubs of multiply perfect numbers (excluding the perfect numbers themselves) are conjectured to be finite.

Asymptotic density

Every pair a, b of friendly numbers gives rise to a positive proportion of all natural numbers being friendly (but in different clubs), by considering pairs na, nb for multipliers n with gdc(n, ab) = 1. For example, the "primitive" friendly pair 6 and 28 gives rise to friendly pairs 6n and 28n for all n that are congruent to 1, 5, 11, 13, 17, 19, 23, 25, 29, 31, 37, or 41 modulo 42.[4]

This shows that the natural density of the friendly numbers (if it exists) is positive.

Anderson and Hickerson proposed that the density should in fact be 1 (or equivalently that the desity of the solitary numbers should be 0).[4]. According to the MathWorld article on Solitary Number (see References section below), this conjecture has not been resolved, although Pomerance thought at one point he had disproved it.

Notes

1. ^ a b Cemra, Jason. "10 Solitary Check". Github/CemraJC/Solidarity.
2. ^ a b OEIS sequence A074902
3. ^ Flammenkamp, Achim. "The Multiply Perfect Numbers Page". Retrieved 2008-04-20.
4. ^ a b Claude W. Anderson and Dean Hickerson, Problem 6020: Friendly Integers, Amer. Math. Monthly 84 (1977) pp. 65-66.