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

30, 60, 70, 78, 102, 110, 120, 140, 150, 156, 174, 182, 190, 204, 210, 220, 222, 230, 238, 240, 246, 270, 280, 286, 300, 310, 312, 318, 330, 348, 350, 364, 366, 374, 380, 390, 406, 408, 420, 430, 438, 440, 444, 460, 470, 476, 480, 492, 494, 510, 518, 534, 540, 546, 550, 560

Offset: 1

Author

Altug Alkan, Jan 22 2016

Keywords

• nonn

Comments

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, ...

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.

Examples

			30 is a term because 30 is not the sum of 2 nonzero squares and 30^30 = 8609344200000000000000^2 + 11479125600000000000000^2.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000312, A000404, A018825, A162592.

Programs

Mathematica

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 *)

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));}
for(n=1, 1e3, if(isA000404(n^n) && !isA000404(n), print1(n, ", ")));