Aliquot sequence classifications

Aliquot sequence classifications is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

An aliquot sequence of a positive integer K is defined recursively as the first member being K and subsequent members being the sum of the Proper divisors of the previous term.

• If the terms eventually reach 0 then the series for K is said to terminate.

There are several classifications for non termination:

• If the second term is K then all future terms are also K and so the sequence repeats from the first term with period 1 and K is called perfect.
• If the third term would be repeating K then the sequence repeats with period 2 and K is called amicable.
• If the N'th term would be repeating K for the first time, with N > 3 then the sequence repeats with period N - 1 and K is called sociable.

Perfect, amicable and sociable numbers eventually repeat the original number K; there are other repetitions...

• Some K have a sequence that eventually forms a periodic repetition of period 1 but of a number other than K, for example 95 which forms the sequence 95, 25, 6, 6, 6, ... such K are called aspiring.
• K that have a sequence that eventually forms a periodic repetition of period >= 2 but of a number other than K, for example 562 which forms the sequence 562, 284, 220, 284, 220, ... such K are called cyclic.

And finally:

• Some K form aliquot sequences that are not known to be either terminating or periodic. these K are to be called non-terminating.
  For the purposes of this task, K is to be classed as non-terminating if it has not been otherwise classed after generating 16 terms or if any term of the sequence is greater than 2**47 = 140737488355328.

Task

1. Create routine(s) to generate the aliquot sequence of a positive integer enough to classify it according to the classifications given above.
2. Use it to display the classification and sequences of the numbers one to ten inclusive.
3. Use it to show the classification and sequences of the following integers, in order:

11, 12, 28, 496, 220, 1184, 12496, 1264460, 790, 909, 562, 1064, 1488, and optionally 15355717786080.

Show all output on this page.

Cf.

• Abundant, deficient and perfect number classifications. (Classifications from only the first two members of the whole sequence).
• Proper divisors
• Amicable pairs

Python

Importing Proper divisors from prime factors:

<lang python>from proper_divisors import proper_divs from functools import lru_cache

@lru_cache() def pdsum(n):

   return sum(proper_divs(n))
   
   

def aliquot(n, maxlen=16, maxterm=2**47):

   if n == 0:
       return 'terminating', [0]
   s, slen, new = [n], 1, n
   while slen <= maxlen and new < maxterm:
       new = pdsum(s[-1])
       if new in s:
           if s[0] == new:
               if slen == 1:
                   return 'perfect', s
               elif slen == 2:
                   return 'amicable', s
               else:
                   return 'sociable of length %i' % slen, s
           elif s[-1] == new:
               return 'aspiring', s
           else:
               return 'cyclic back to %i' % new, s
       elif new == 0:
           return 'terminating', s + [0]
       else:
           s.append(new)
           slen += 1
   else:
       return 'non-terminating', s
               

if __name__ == '__main__':

   for n in range(1, 11): 
       print('%s: %r' % aliquot(n))
   print()
   for n in [11, 12, 28, 496, 220, 1184,  12496, 1264460, 790, 909, 562, 1064, 1488, 15355717786080]: 
       print('%s: %r' % aliquot(n))</lang>

Output:

terminating: [1, 0]
terminating: [2, 1, 0]
terminating: [3, 1, 0]
terminating: [4, 3, 1, 0]
terminating: [5, 1, 0]
perfect: [6]
terminating: [7, 1, 0]
terminating: [8, 7, 1, 0]
terminating: [9, 4, 3, 1, 0]
terminating: [10, 8, 7, 1, 0]

terminating: [11, 1, 0]
terminating: [12, 16, 15, 9, 4, 3, 1, 0]
perfect: [28]
perfect: [496]
amicable: [220, 284]
amicable: [1184, 1210]
sociable of length 5: [12496, 14288, 15472, 14536, 14264]
sociable of length 4: [1264460, 1547860, 1727636, 1305184]
aspiring: [790, 650, 652, 496]
aspiring: [909, 417, 143, 25, 6]
cyclic back to 284: [562, 284, 220]
cyclic back to 1184: [1064, 1336, 1184, 1210]
non-terminating: [1488, 2480, 3472, 4464, 8432, 9424, 10416, 21328, 22320, 55056, 95728, 96720, 236592, 459792, 881392, 882384, 1474608]
non-terminating: [15355717786080, 44534663601120, 144940087464480]