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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
nσ (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
nσ (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
nσ (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
nσ (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
    )



External links