Amicable pairs
Two integers N and M are said to be amicable pairs if N != M and the sum of the proper divisors of N == M as well as the sum of the proper divisors of M == N.
For example 1184 and 1210 are an amicable pair (with proper divisors 1, 2, 4, 8, 16, 32, 37, 74, 148, 296, 592 and 1, 2, 5, 10, 11, 22, 55, 110, 121, 242, 605 respectively).
- Task
Calculate and show here the Amicable pairs below 20,000; (there are eight).
- Cf.
J
Proper Divisor implementation:
<lang J>factors=: [: /:~@, */&>@{@((^ i.@>:)&.>/)@q:~&__ properDivisors=: factors -. -.&1</lang>
Amicable pairs:
<lang J> 1+0 20000 #:I.,(</~@i.@#*(*|:))(=/ +/@properDivisors@>) 1+i.20000
220 284 1184 1210 2620 2924 5020 5564 6232 6368
10744 10856 12285 14595 17296 18416</lang>
Python
Importing Proper divisors from prime factors: <lang python>from proper_divisors import proper_divs
def amicable(rangemax=20000):
n2divsum = {n: sum(proper_divs(n)) for n in range(1, rangemax + 1)} for num, divsum in n2divsum.items(): if num < divsum and divsum <= rangemax and n2divsum[divsum] == num: yield num, divsum
if __name__ == '__main__':
for num, divsum in amicable(): print('Amicable pair: %i and %i With proper divisors:\n %r\n %r' % (num, divsum, sorted(proper_divs(num)), sorted(proper_divs(divsum))))</lang>
- Output:
Amicable pair: 220 and 284 With proper divisors: [1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110] [1, 2, 4, 71, 142] Amicable pair: 1184 and 1210 With proper divisors: [1, 2, 4, 8, 16, 32, 37, 74, 148, 296, 592] [1, 2, 5, 10, 11, 22, 55, 110, 121, 242, 605] Amicable pair: 2620 and 2924 With proper divisors: [1, 2, 4, 5, 10, 20, 131, 262, 524, 655, 1310] [1, 2, 4, 17, 34, 43, 68, 86, 172, 731, 1462] Amicable pair: 5020 and 5564 With proper divisors: [1, 2, 4, 5, 10, 20, 251, 502, 1004, 1255, 2510] [1, 2, 4, 13, 26, 52, 107, 214, 428, 1391, 2782] Amicable pair: 6232 and 6368 With proper divisors: [1, 2, 4, 8, 19, 38, 41, 76, 82, 152, 164, 328, 779, 1558, 3116] [1, 2, 4, 8, 16, 32, 199, 398, 796, 1592, 3184] Amicable pair: 10744 and 10856 With proper divisors: [1, 2, 4, 8, 17, 34, 68, 79, 136, 158, 316, 632, 1343, 2686, 5372] [1, 2, 4, 8, 23, 46, 59, 92, 118, 184, 236, 472, 1357, 2714, 5428] Amicable pair: 12285 and 14595 With proper divisors: [1, 3, 5, 7, 9, 13, 15, 21, 27, 35, 39, 45, 63, 65, 91, 105, 117, 135, 189, 195, 273, 315, 351, 455, 585, 819, 945, 1365, 1755, 2457, 4095] [1, 3, 5, 7, 15, 21, 35, 105, 139, 417, 695, 973, 2085, 2919, 4865] Amicable pair: 17296 and 18416 With proper divisors: [1, 2, 4, 8, 16, 23, 46, 47, 92, 94, 184, 188, 368, 376, 752, 1081, 2162, 4324, 8648] [1, 2, 4, 8, 16, 1151, 2302, 4604, 9208]