A262091 - OEIS

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.

%I #27 Jan 09 2016 14:10:13

%S 136,919,2178,58618,89883,63804,2755907,8139850,144839908,277668893,

%T 304162700,4370652168,21914086555935085,187864919457180831,

%U 13397885590701080090,19095442247273220984552,108493282045082839040458,1553298727699254868304830

%N 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.

%C 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

%H D. Knuth, <a href="http://www-cs-faculty.stanford.edu/~knuth/amicable_digit_powers.txt">Table of a(n) and its mate for n=1..36</a>

%H K. Oséki, <a href="http://dx.doi.org/10.3792/pja/1195523903">A problem of number theory</a>, Proceedings of the Japan Academy 36 (1960), 578-587.

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

%o (Python)

%o # print pairs with leading zeros

%o from __future__ import print_function

%o from itertools import combinations_with_replacement

%o for m in range(2,11):

%o fs = '0'+str(m+1)+'d'

%o for c in combinations_with_replacement(range(10),m+1):

%o n = sum(d**m for d in c)

%o r = sum(int(q)**m for q in str(n))

%o rlist = sorted(int(d) for d in str(r))

%o rlist = [0]*(m+1-len(rlist))+rlist

%o if n < r and rlist == list(c):

%o print(format(n,fs),format(r,fs)) # _Chai Wah Wu_, Jan 04 2016

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

%K nonn,base

%O 1,1

%A _Don Knuth_, Sep 10 2015

%E Definition clarified by _Chai Wah Wu_, Jan 04 2016