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.
- 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
- 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.
- Some K form aliquot sequences that are not known to be either terminating or periodic. these K are to be called non-terminating.
- Task
- Create routine(s) to generate the aliquot sequence of a positive integer enough to classify it according to the classifications given above.
- Use it to display the classification and sequences of the numbers one to ten inclusive.
- Use it to show the classification and sequences of the following integers:
- 11, 12, 28, 496, 220, 1184, 12496, 1264460, 790, 909, 562, 1064, 1488, 15355717786080
Show all output on this page.
- Cf.
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]