Friendly number

Last updated January 15, 2025

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.

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.

Examples

As another example, 30 and 140 form a friendly pair, because 30 and 140 have the same abundancy:^[1]

{\dfrac {\sigma (30)}{30}}={\dfrac {1+2+3+5+6+10+15+30}{30}}={\dfrac {72}{30}}={\dfrac {12}{5}}

{\dfrac {\sigma (140)}{140}}={\dfrac {1+2+4+5+7+10+14+20+28+35+70+140}{140}}={\dfrac {336}{140}}={\dfrac {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 (deficient)). There are also cases of even numbers being friendly to odd numbers, such as 42, 3472, 56896, ... (sequence A347169 in the OEIS ) and 544635 (abundancy of 16/7). The odd friend may be less than the even one, as in 84729645 and 155315394 (abundancy of 896/351), or in 6517665, 14705145 and 2746713837618 (abundancy of 64/27).

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

Status for small n

In the table below, blue numbers are proven friendly (sequence A074902 in the OEIS ), red numbers are proven solitary (sequence A095739 in the OEIS ), numbers n such that n and $\sigma (n)$ are coprime (sequence A014567 in the OEIS ) are left uncolored, though they are known to be solitary. Other numbers have unknown status and are yellow.

$n$	$\sigma (n)$	${\frac {\sigma (n)}{n}}$
1	1	1
2	3	3/2
3	4	4/3
4	7	7/4
5	6	6/5
6	12	2
7	8	8/7
8	15	15/8
9	13	13/9
10	18	9/5
11	12	12/11
12	28	7/3
13	14	14/13
14	24	12/7
15	24	8/5
16	31	31/16
17	18	18/17
18	39	13/6
19	20	20/19
20	42	21/10
21	32	32/21
22	36	18/11
23	24	24/23
24	60	5/2
25	31	31/25
26	42	21/13
27	40	40/27
28	56	2
29	30	30/29
30	72	12/5
31	32	32/31
32	63	63/32
33	48	16/11
34	54	27/17
35	48	48/35
36	91	91/36

$n$	$\sigma (n)$	${\frac {\sigma (n)}{n}}$
37	38	38/37
38	60	30/19
39	56	56/39
40	90	9/4
41	42	42/41
42	96	16/7
43	44	44/43
44	84	21/11
45	78	26/15
46	72	36/23
47	48	48/47
48	124	31/12
49	57	57/49
50	93	93/50
51	72	24/17
52	98	49/26
53	54	54/53
54	120	20/9
55	72	72/55
56	120	15/7
57	80	80/57
58	90	45/29
59	60	60/59
60	168	14/5
61	62	62/61
62	96	48/31
63	104	104/63
64	127	127/64
65	84	84/65
66	144	24/11
67	68	68/67
68	126	63/34
69	96	32/23
70	144	72/35
71	72	72/71
72	195	65/24

$n$	$\sigma (n)$	${\frac {\sigma (n)}{n}}$
73	74	74/73
74	114	57/37
75	124	124/75
76	140	35/19
77	96	96/77
78	168	28/13
79	80	80/79
80	186	93/40
81	121	121/81
82	126	63/41
83	84	84/83
84	224	8/3
85	108	108/85
86	132	66/43
87	120	40/29
88	180	45/22
89	90	90/89
90	234	13/5
91	112	16/13
92	168	42/23
93	128	128/93
94	144	72/47
95	120	24/19
96	252	21/8
97	98	98/97
98	171	171/98
99	156	52/33
100	217	217/100
101	102	102/101
102	216	36/17
103	104	104/103
104	210	105/52
105	192	64/35
106	162	81/53
107	108	108/107
108	280	70/27

$n$	$\sigma (n)$	${\frac {\sigma (n)}{n}}$
109	110	110/109
110	216	108/55
111	152	152/111
112	248	31/14
113	114	114/113
114	240	40/19
115	144	144/115
116	210	105/58
117	182	14/9
118	180	90/59
119	144	144/119
120	360	3
121	133	133/121
122	186	93/61
123	168	56/41
124	224	56/31
125	156	156/125
126	312	52/21
127	128	128/127
128	255	255/128
129	176	176/129
130	252	126/65
131	132	132/131
132	336	28/11
133	160	160/133
134	204	102/67
135	240	16/9
136	270	135/68
137	138	138/137
138	288	48/23
139	140	140/139
140	336	12/5
141	192	64/47
142	216	108/71
143	168	168/143
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.

Is 10 a solitary number?

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 $10^{30}$ .^[2]^[3] J.Ward ^[4]proved that any positive integer $n$ other than 10 with abundancy index 9/5 must be a square with at least six distinct prime factors, the smallest being 5. Further, at least one of the prime factors must be congruent to 1 modulo 3 and appear with an exponent congruent to 2 modulo 6 in the prime power factorization of $n$ . In ^[5] the authors proposed necessary upper bounds for the second, third and fourth smallest prime divisors of friends of 10, if $n$ is a friend of 10 and if $q_{2},q_{3},q_{4}$ are the second, third, fourth smallest prime divisors of $n$ respectively, then

$7\leq q_{2}<\left\lceil {\frac {7\omega (n)}{3}}\right\rceil {\biggl (}\log \left\lceil {\frac {7\omega (n)}{3}}\right\rceil +2\log \log \left\lceil {\frac {7\omega (n)}{3}}\right\rceil {\biggr )},$

$11\leq q_{3}<\left\lceil {\frac {180\omega (n)}{41}}\right\rceil {\biggl (}\log \left\lceil {\frac {180\omega (n)}{41}}\right\rceil +2\log \log \left\lceil {\frac {180\omega (n)}{41}}\right\rceil {\biggr )},$

$13\leq q_{4}<\left\lceil {\frac {390\omega (n)}{47}}\right\rceil {\biggl (}\log \left\lceil {\frac {390\omega (n)}{47}}\right\rceil +2\log \log \left\lceil {\frac {390\omega (n)}{47}}\right\rceil {\biggr )},$

where $\omega (n)$ is the number of distinct prime divisors of $n$ and $\left\lceil .\right\rceil$ is the ceiling function. Further, they proposed reasonable upper bounds for all prime divisors of friends of 10 in ^[6], for any smallest r-th prime divisor $q_{r}$ of $n$ , we have

$q_{r}<L(\log L+2\log \log L),$ where

$L=\left\lceil {\frac {{\mathcal {A}}\omega (n)}{\mathcal {B}}}\right\rceil$ , ${\frac {\mathcal {A}}{\mathcal {B}}}>{\frac {1}{{\frac {36}{25}}\prod _{i=4}^{r+1}{\bigl (}1-{\frac {1}{p_{i}}}{\bigr )}-1}}$ ,

where $p_{i}$ is the i-th prime number and ${\mathcal {A,B}}\in \mathbb {Z} ^{+}$ .

Small numbers with a relatively large smallest friend do exist: for instance, 24 is friendly, with its smallest friend 91,963,648.^[2]^[3]

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. There are clubs with more known members: in particular, those formed by multiply perfect numbers, which are numbers whose abundancy is an integer. 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 gcd(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.^[7]

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 density of the solitary numbers should be 0).^[7] 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

↑ "Numbers with Cool Names: Amicable, Sociable, Friendly". 10 May 2023. Retrieved 26 July 2023.
1 2 Cemra, Jason (23 July 2022). "10 Solitary Check". Github/CemraJC/Solidarity.
1 2 "OEIS sequence A074902". On-Line Encyclopedia of Integer Sequences . Retrieved 10 July 2020.
↑ Ward, Jeffrey (2008-06-06), Does Ten Have a Friend?, doi:10.48550/arXiv.0806.1001 , retrieved 2025-01-14
↑ Mandal, Sourav; Mandal, Sagar (2024-11-04), Upper bounds for the prime divisors of friends of 10, doi:10.48550/arXiv.2404.05771 , retrieved 2025-01-14
↑ Mandal, Sagar; Mandal, Sourav (2024-11-10), Upper bounds for the prime divisors of friends of 10-II, doi:10.48550/arXiv.2412.02701 , retrieved 2025-01-15
1 2 Anderson, C. W.; Hickerson, Dean; Greening, M. G. (1977). "6020". The American Mathematical Monthly. 84 (1): 65–66. doi:10.2307/2318325. JSTOR 2318325.

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References

Grime, James. A video about the number 10. Numberphile .
Weisstein, Eric W. "Friendly Number". MathWorld .
Weisstein, Eric W. "Friendly Pair". MathWorld .
Weisstein, Eric W. "Solitary Number". MathWorld .
Weisstein, Eric W. "Abundancy". MathWorld .

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[1] "Numbers with Cool Names: Amicable, Sociable, Friendly". 10 May 2023. Retrieved 26 July 2023.

[:0-2] 1 2 Cemra, Jason (23 July 2022). "10 Solitary Check". Github/CemraJC/Solidarity.

[:1-3] 1 2 "OEIS sequence A074902". On-Line Encyclopedia of Integer Sequences . Retrieved 10 July 2020.

[4] Ward, Jeffrey (2008-06-06), Does Ten Have a Friend?, doi:10.48550/arXiv.0806.1001 , retrieved 2025-01-14

[5] Mandal, Sourav; Mandal, Sagar (2024-11-04), Upper bounds for the prime divisors of friends of 10, doi:10.48550/arXiv.2404.05771 , retrieved 2025-01-14

[6] Mandal, Sagar; Mandal, Sourav (2024-11-10), Upper bounds for the prime divisors of friends of 10-II, doi:10.48550/arXiv.2412.02701 , retrieved 2025-01-15

[AndersonHickerson-7] 1 2 Anderson, C. W.; Hickerson, Dean; Greening, M. G. (1977). "6020". The American Mathematical Monthly. 84 (1): 65–66. doi:10.2307/2318325. JSTOR 2318325.

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v t e Divisibility-based sets of integers
Overview	Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic
Factorization forms	Prime Composite Semiprime Pronic Sphenic Square-free Powerful Perfect power Achilles Smooth Regular Rough Unusual
Constrained divisor sums	Perfect Almost perfect Quasiperfect Multiply perfect Hemiperfect Hyperperfect Superperfect Unitary perfect Semiperfect Practical Descartes Erdős–Nicolas
With many divisors	Abundant Primitive abundant Highly abundant Superabundant Colossally abundant Highly composite Superior highly composite Weird
Aliquot sequence-related	Untouchable Amicable (Triple) Sociable Betrothed
Base-dependent	Equidigital Extravagant Frugal Harshad Polydivisible Smith
Other sets	Arithmetic Deficient Friendly Solitary Sublime Harmonic divisor Refactorable Superperfect