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A267967 Integers n such that n^n is the sum of two nonzero squares while n is not.

%I A267967 #23 Mar 15 2016 00:10:13
%S A267967 30,60,70,78,102,110,120,140,150,156,174,182,190,204,210,220,222,230,
%T A267967 238,240,246,270,280,286,300,310,312,318,330,348,350,364,366,374,380,
%U A267967 390,406,408,420,430,438,440,444,460,470,476,480,492,494,510,518,534,540,546,550,560
%N A267967 Integers n such that n^n is the sum of two nonzero squares while n is not.
%C A267967 Terms that are not divisible by 10 are 78, 102, 156, 174, 182, 204, 222, 238, 246, 286, 312, 318, 348, 364, 366, 374, 406, 408, 438, 444, 476, 492, 494, 518, 534, 546, 572, 574, 582, 598, 606, 624, 636, 638, 646, 654, 678, 696, 728, ...
%C A267967 If k^2 is the sum of 2 nonzero squares, (2*k)^(2*k) is. (2*k)^(2*k) = 2^(2*k) * k^(2*k) = (2^k)^2 * k^2 * k^(2*k-2) = (2^k)^2 * k^2 * (k^(k-1))^2. So if k^2 = a^2 + b^2, then (2*k)^(2*k) = (k^(k-1)*2^k*a)^2 + (k^(k-1)*2^k*b)^2. And if k^2 = a^2 + b^2 and k is not the sum of 2 nonzero squares, 2*k is not the sum of 2 nonzero squares. So 2 * A162592(n) appears in this sequence. Note that all terms appear as even numbers.
%H A267967 Chai Wah Wu, <a href="/A267967/b267967.txt">Table of n, a(n) for n = 1..10000</a>
%e A267967 30 is a term because 30 is not the sum of 2 nonzero squares and 30^30 = 8609344200000000000000^2 + 11479125600000000000000^2.
%t A267967 Select[Range@ 120, SquaresR[2, #] == 0 && Resolve[Exists[{a, b}, Reduce[#^# == (a^2 + b^2), {a, b}, Integers], a > b > 0]] &] (* _Michael De Vlieger_, Jan 24 2016 *)
%o A267967 (PARI) isA000404(n) = {for( i=1, #n=factor(n)~%4, n[1, i]==3 && n[2, i]%2 && return); n && ( vecmin(n[1, ])==1 || (n[1, 1]==2 && n[2, 1]%2));}
%o A267967 for(n=1, 1e3, if(isA000404(n^n) && !isA000404(n), print1(n, ", ")));
%Y A267967 Cf. A000312, A000404, A018825, A162592.
%K A267967 nonn
%O A267967 1,1
%A A267967 _Altug Alkan_, Jan 22 2016