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%I #9 Mar 15 2022 11:08:30
%S 7,18,45,61,72,85,90,145,270,306,315,367,376,448,477,540,547,585,667,
%T 733,756,765,943,1152,1377,1899,1971,2106,2133,2155,2215,2224,2349,
%U 2628,2822,2871,2968,3123,3139,3181,3204,3355,3546,3553,3775,3780,4131,4455
%N Numbers n that can be expressed as the sum of the digits of both m^k and k^m for distinct numbers m and k which are not both equal to powers of 10.
%C If "sumdigit" denotes the sum of the digits of a number then these are the numbers n such that n=sumdigit(m^k)=sumdigit(k^m).
%C Two banal cases are not considered: 1) m=k because m^k=k^m and the sum of the digits is automatically equal for both the numbers; 2) powers of 10 because sumdigit(10^a)=1 for any integer a. The same number can be generated by different pairs: 477 cames from sumdigit(54^63)=sumdigit(63^54) and sumdigit(90^120)=sumdigit(120^90) 2349 cames from sumdigit(216^222)=sumdigit(222^216), sumdigit(216^225)=sumdigit(225^216) and sumdigit(219^222)=sumdigit(222^219)
%e 270 = sumdigit(36^39) = sumdigit(39^36);
%e 1152 = sumdigit(114^126) = sumdigit(126^114);
%e 2133 = sumdigit(204^213) = sumdigit(213^204).
%p P:=proc(n)local i,j,k,w,x,y; for i from 1 by 1 to n do for j from 1 by 1 to n do w:=0; x:=0; k:=i^j; y:=j^i; while k>0 do w:=w+k-trunc(k/10)*10; k:=trunc(k/10); od; while y>0 do x:=x+y-trunc(y/10)*10; y:=trunc(y/10); od; if (w=x) and (w<>1) and (i<j) then print(i,j,w); fi; od; od; end: P(500);
%Y Cf. A124359, A124360, A046019, A124365, A124366, A124367.
%K nonn,base
%O 1,1
%A _Paolo P. Lava_ and _Giorgio Balzarotti_, Nov 03 2006, Nov 29 2006
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