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A268511 Odd integers n such that 3^n + 5^n = x^2 + y^2 (x and y integers) is solvable.

%I A268511 #29 Oct 04 2021 13:13:39
%S A268511 1,5,13,17,29,89,109,149,157,193,373
%N A268511 Odd integers n such that 3^n + 5^n = x^2 + y^2 (x and y integers) is solvable.
%C A268511 Corresponding 3^n + 5^n values are 8, 3368, 1222297448, 763068593288, 186264583553473068008, ...
%C A268511 445 <= a(12) <= 509. 509, 661, 709 are terms. - _Chai Wah Wu_, Jul 22 2020
%e A268511 1 is a term because 3^1 + 5^1 = 8 = 2^2 + 2^2.
%e A268511 5 is a term because 3^5 + 5^5 = 3368 = 2^2 + 58^2.
%e A268511 13 is a term because 3^13 + 5^13 = 1222297448 = 4118^2 + 34718^2.
%t A268511 Select[Range[1, 110, 2], Resolve@ Exists[{x, y}, Reduce[3^# + 5^# == (x^2 + y^2), {x, y}, Integers]] &] (* _Michael De Vlieger_, Feb 07 2016 *)
%o A268511 (PARI) is(n) = #bnfisintnorm(bnfinit(z^2+1), n);
%o A268511 for(n=1, 1e3, if(n%2==1 && is(3^n + 5^n), print1(n, ", ")));
%o A268511 (Python)
%o A268511 from sympy import factorint
%o A268511 A268511_list = []
%o A268511 for n in range(1,50,2):
%o A268511     m = factorint(3**n+5**n)
%o A268511     for d in m:
%o A268511         if d % 4 == 3 and m[d] % 2:
%o A268511             break
%o A268511     else:
%o A268511         A268511_list.append(n) # _Chai Wah Wu_, Dec 26 2018
%Y A268511 Cf. A001481, A074606.
%K A268511 nonn,more
%O A268511 1,2
%A A268511 _Altug Alkan_, Feb 06 2016
%E A268511 a(8)-a(9) from _Giovanni Resta_, Apr 10 2016
%E A268511 a(10)-a(11) from _Chai Wah Wu_, Jul 22 2020