A262091 Amicable digital pairs: The smaller number of a pair (x,y) with x <> y such that, in decimal notation and with an appropriate number of leading zeros prepended, x=(x_m...x_1x_0){10}, y=(y_m...y_1y_0){10}, x = y_m^m + ... + y_1^m + y_0^m, and y = x_m^m + ... + x_1^m + x_0^m.

136, 919, 2178, 58618, 89883, 63804, 2755907, 8139850, 144839908, 277668893, 304162700, 4370652168, 21914086555935085, 187864919457180831, 13397885590701080090, 19095442247273220984552, 108493282045082839040458, 1553298727699254868304830

Offset: 1

Author

Don Knuth, Sep 10 2015

Keywords

• nonn
• base

Comments

If we allow x to be equal to y we get numbers such as 1, 153, 370, 371, 407, ... See A252648. - Chai Wah Wu, Jan 04 2016

Examples

			a(1) is amicably paired to 244, because 1^3 + 3^3 + 6^3 = 244 and 2^3 + 4^3 + 4^3 = 136.
		

Links

• D. Knuth, Table of a(n) and its mate for n=1..36
• K. Oséki, A problem of number theory, Proceedings of the Japan Academy 36 (1960), 578-587.

Crossrefs

A262092 has the larger element of each pair. Cf. A252648.

Programs

Python

# print pairs with leading zeros
from _future_ import print_function
from itertools import combinations_with_replacement
for m in range(2,11):
    fs = '0'+str(m+1)+'d'
    for c in combinations_with_replacement(range(10),m+1):
        n = sum(d**m for d in c)
        r = sum(int(q)**m for q in str(n))
        rlist = sorted(int(d) for d in str(r))
        rlist = [0]*(m+1-len(rlist))+rlist
        if n < r and rlist == list(c):
            print(format(n,fs),format(r,fs)) # Chai Wah Wu, Jan 04 2016

Extensions

Definition clarified by Chai Wah Wu, Jan 04 2016