16 + 32 + 40787 + 81574 + 163148 + 326296 + 652592 = 1264460. List of known sociable numbers The following categorizes all known sociable numbers by the length of the corresponding aliquot sequence: It is conjectured that if n is congruent to 3 modulo 4 then there is no such sequence with length n. The 5-cycle sequence is: 12496, 14288, 15472, 14536, 14264 The only known 28-cycle is: 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 . These two sequences provide the only sociable numbers below 1 million (other than the perfect and amicable numbers). Searching for sociable numbers The aliquot sequence can be represented as a directed graph, , for a given integer , where denotes the sum of the proper divisors of . Cycles in represent sociable numbers within the interval . Two special cases are loops that represent perfect numbers and cycles of length two that represent amicable pairs. Conjecture of the sum of sociable number cycles It is conjectured that as the number of sociable number cycles with length greater than 2 approaches infinity, the proportion of the sums of the sociable number cycles divisible by 10 approaches 1 . References H. Cohen, On amicable and sociable numbers, Math. Comp. 24 (1970), pp. 423–429 External links A list of known sociable numbers Extensive tables of perfect, amicable and sociable numbers A003416 (smallest sociable number from each cycle) and A122726 (all sociable numbers) in OEIS Arithmetic dynamics Divisor function Integer sequences Number theory