Sociable numbers

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Integers

n0, ..., nk  − 1

are sociable numbers (a

k

-tuple of sociable numbers of order

k

 ) if

s (ni )  :=  σ (ni ) − ni  =  n(i + 1 mod k ) , i = 0 .. k − 1, k ≥ 3,

where

s (n)

is the sum of aliquot divisors of

n

and

σ (n)

is the sum of divisors of

n

. One might say that a sociable

k

-tuple of numbers is mutually perfect (so to speak) since

k  − 1 ∑   i  = 0    σ (ni ) − 2   k  − 1 ∑   i  = 0    ni  =  0.

The sociable

k

-tuples (

k   ≥   3

) are

{(12496, 14288, 15472, 14536, 14264, 12496), (14316, 19116, 31704, 47616, 83328, 177792, 295488, 629072, 589786, 294896, 358336, 418904, 366556, 274924, 275444, 243760, 376736, 381028, 285778, 152990, 122410, 97946, 48976, 45946, 22976, 22744, 19916, 17716, 14316), (1264460, 1547860, 1727636, 1305184, 1264460), ...}

Amicable numbers ([not really] sociable numbers of order 2) correspond to

k = 2

, while perfect numbers ([really not] sociable numbers of order 1) correspond to

k = 1

.

There are no known sociable numbers of order 3 (do they exist?).

Example

{{#expr: {{divisor function|12496}} - 12496 }} = 14288,

{{#expr: {{divisor function|14288}} - 14288 }} = 15472,

{{#expr: {{divisor function|15472}} - 15472 }} = 14536,

{{#expr: {{divisor function|14536}} - 14536 }} = 14264,

{{#expr: {{divisor function|14264}} - 14264 }} = 12496,

{{#expr: {{divisor function|12496}} - 12496 }} = 14288.

Sequences

A003416 Sociable numbers: smallest member of each cycle.

{12496, 14316, 1264460, 2115324, 2784580, 4938136, 7169104, 18048976, 18656380, 28158165, 46722700, 81128632, 174277820, 209524210, 330003580, 498215416, 805984760, 1095447416, 1236402232, 1276254780, 1799281330, ...}

A072891 The 5-cycle of the

n ⤇ σ (n)  −  n

process starting at 12496, where

σ (n)

is the sum of divisors of

n

(A000203).

{12496, 14288, 15472, 14536, 14264, 12496}

A072890 The 28-cycle of the

n ⤇ σ (n)  −  n

process starting at 14316, where

σ (n)

is the sum of divisors of

n

(A000203).

{14316, 19116, 31704, 47616, 83328, 177792, 295488, 629072, 589786, 294896, 358336, 418904, 366556, 274924, 275444, 243760, 376736, 381028, 285778, 152990, 122410, 97946, 48976, 45946, 22976, 22744, 19916, 17716, 14316}

A072892 The 4-cycle of the

n ⤇ σ (n)  −  n

process starting at 1264460, where

σ (n)

is the sum of divisors of

n

. (A000203).

{1264460, 1547860, 1727636, 1305184, 1264460}

A?????? The 4-cycle of the

n ⤇ σ (n)  −  n

process starting at 28158165, where

σ (n)

is the sum of divisors of

n

. (A000203).

{28158165, 29902635, 30853845, 29971755, 28158165}

A090615 Smallest member of sociable quadruples.

{1264460, 2115324, 2784580, 4938136, 7169104, 18048976, 18656380, 28158165, 46722700, 81128632, 174277820, 209524210, 330003580, 498215416, 1236402232, 1799281330, 2387776550, 2717495235, 2879697304, 3705771825, 4424606020, ...}

A?????? Smallest member of sociable quintuples.

{12496, ...}

A?????? Smallest member of sociable 28-tuples.

{14316, ...}

See also

• Perfect numbers (singles) (

  σ (n)  −  n = n

  )
• Amicable numbers (pairs) (

  σ (m)  −  m = n

  and

  σ (n)  −  n = m

  )
• Sociable numbers ( 

  k

  -tuples) (

  σ (ni )  −  ni = n(i +1 mod k ) , i = 0 .. k  −  1, k   ≥   3

  )

• {{divisor function}} (arithmetic function template)

External links

• A list of known sociable numbers
• Paul Pollack (joint with Mits Kobayashi and Carl Pomerance), The distribution of sociable numbers, 2009.