%I #30 Apr 17 2021 02:16:37

%S 4,16,196,2836,5956,25936,65536,540736,598816,797476,1151536,3704416,

%T 8095984,11272276,13362420,21235696,29640832,31084096,42913396,

%U 49960912,55137316,70254724,70836676,81158416,94618996,111849956,129275056,150026176,168267856,169242676,189796420,192226516,198464176,208232116,244553296,246605776,300018016,318143296

%N Even numbers k such that k + 2 divides k^k + 2.

%C In other words, even numbers k such that k + 2 divides A014566(k) + 1.

%C Even terms of A213382.

%C 4, 16, 65536 are the numbers of the form 2^(2^(2^k)), for k >= 0. Are there other members of this sequence with the form of 2^(2^(2^k))?

%C 2^(2^(2^3)) and 2^(2^(2^4)) are terms. - _Michael S. Branicky_, Apr 16 2021

%H Michael S. Branicky, <a href="/A271267/b271267.txt">Table of n, a(n) for n = 1..150</a>

%e 4 is a term because 4 + 2 = 6 divides 4^4 + 2 = 258.

%t Select[Range[2, 10^4, 2], Divisible[#^# + 2, # + 2] &] (* _Michael De Vlieger_, Apr 03 2016 *)

%o (PARI) lista(nn) = forstep(n=2, nn, 2, if( Mod(n, n+2)^n == -2 , print1(n, ", "))); \\ _Joerg Arndt_, Apr 03 2016

%o (Python)

%o def afind(limit):

%o k = 2

%o while k < limit:

%o if (pow(k, k, k+2) + 2)%(k+2) == 0: print(k, end=", ")

%o k += 2

%o afind(10**7) # _Michael S. Branicky_, Apr 16 2021

%Y Cf. A014566, A081765, A213382.

%K nonn

%O 1,1

%A _Altug Alkan_, Apr 03 2016