Friendly number

Demonstration, with Cuisenaire rods, of the abundance of the number 12

In number theory, friendly numbers are two or more natural numbers with a common abundancy, 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 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 m ≠ n such that σ(m) / m = σ(n) / n. Note that abundancy is not the same as abundance which is defined as σ(n) − 2n.

Abundancy may also be expressed as $\sigma_{-\!1}(n)$ where $\sigma_k$ denotes a divisor function with $\sigma_{k}(n)$ 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. 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.

Example

Blue numbers are proved friendly (sequence A074902 in OEIS), red numbers are proved solitary, numbers n such that n and $\sigma(n)$ are coprime (sequence A014567 in OEIS) are not coloured here, though they are known to be solitary, other numbers have unknown status.

n	$\sigma(n)$	$\frac{\sigma(n)}{n}$	n	$\sigma(n)$	$\frac{\sigma(n)}{n}$	n	$\sigma(n)$	$\frac{\sigma(n)}{n}$	n	$\sigma(n)$	$\frac{\sigma(n)}{n}$
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

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

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

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

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

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 OEIS). For a prime number p we have σ(p) = p + 1, which is coprime with p.

No general method is known for determining whether a number is friendly or solitary. The smallest number whose classification is unknown (as of 2009) is 10; it is conjectured to be solitary; if not, its smallest friend is a fairly large number, like the status for the number 24, although 24 is friendly, its smallest friend is 91,963,648.

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 February 2016, 49 perfect numbers are known, the largest of which has more than 44 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.^[1] Although some are known to be quite large, clubs of multiply perfect numbers (excluding the perfect numbers themselves) are conjectured to be finite.

Notes

1. Flammenkamp, Achim. "The Multiply Perfect Numbers Page". Retrieved 2008-04-20.

References

• Weisstein, Eric W., "Friendly Number", MathWorld.
• Weisstein, Eric W., "Friendly Pair", MathWorld.
• Weisstein, Eric W., "Solitary Number", MathWorld.
• Weisstein, Eric W., "Abundancy", MathWorld.