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%I #10 May 12 2015 17:39:14
%S 63,103,128,147,155,212,272,292,351,452,486,497,525,527,537,584,607,
%T 624,648,729,979,999,1024,1296,1323,1359,1533,1541,1575,1809,1872,
%U 2048,2050,2107,2187,2448,2512,2537,2564,2763,2793,2886,3072,3357,3927,4096,4263,4284
%N Numbers n such that n = concat(a,b) and n | a^b + b^a , with a>0 and b>0.
%C We can have different solutions for the same number. E.g.: 2048 divides both (20^48 + 48^20) and (204^8 + 8^204). The same occurs for 4096, 4263, 16807, 32768, 96957, 156672, 186624, 252081, 262144, 270729, 352947, 390624 … The first number with 3 different concatenations is 186624 that divides (18^6624 + 6624^18), (186^624 + 624^186) and (1866^24 + 24^1866).
%H Paolo P. Lava, <a href="/A257897/b257897.txt">Table of n, a(n) for n = 1..400</a>
%e 6^3 + 3^6 = 945 and 945 / 63 = 15;
%e 10^3 + 3^10 = 60049 and 60049 / 103 = 583;
%e 12^8 + 8^12 = 69149458432 and 69149458432 / 128 = 540230144; etc.
%p with(numtheory); P:=proc(q) local a,b,i,n; for n from 1 to q do
%p for i from 1 to ilog10(n) do a:=trunc(n/10^i); b:=n-a*10^i;
%p if a>0 and b>0 then if type((a^b+b^a)/n,integer)
%p then print(n); break; fi; fi; od; od; end: P(10^9);
%t tst[n_]:=Catch@Block[{a,b}, Do[a=Floor[n/10^k]; b=Mod[n,10^k]; If[Mod[ PowerMod[a, b, n] + PowerMod[b, a, n], n]==0, Throw@True], {k, IntegerLength[n]-1}]; False]; Select[Range@1000, tst] (* _Giovanni Resta_, May 12 2015 *)
%Y Cf. A253824, A253825, A255725, A255726.
%K nonn,base
%O 1,1
%A _Paolo P. Lava_, May 12 2015
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