A268511 - OEIS

%I #29 Oct 04 2021 13:13:39

%S 1,5,13,17,29,89,109,149,157,193,373

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

%C Corresponding 3^n + 5^n values are 8, 3368, 1222297448, 763068593288, 186264583553473068008, ...

%C 445 <= a(12) <= 509. 509, 661, 709 are terms. - _Chai Wah Wu_, Jul 22 2020

%e 1 is a term because 3^1 + 5^1 = 8 = 2^2 + 2^2.

%e 5 is a term because 3^5 + 5^5 = 3368 = 2^2 + 58^2.

%e 13 is a term because 3^13 + 5^13 = 1222297448 = 4118^2 + 34718^2.

%t 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 (PARI) is(n) = #bnfisintnorm(bnfinit(z^2+1), n);

%o for(n=1, 1e3, if(n%2==1 && is(3^n + 5^n), print1(n, ", ")));

%o (Python)

%o from sympy import factorint

%o A268511_list = []

%o for n in range(1,50,2):

%o     m = factorint(3**n+5**n)

%o     for d in m:

%o         if d % 4 == 3 and m[d] % 2:

%o             break

%o     else:

%o         A268511_list.append(n) # _Chai Wah Wu_, Dec 26 2018

%Y Cf. A001481, A074606.

%K nonn,more

%O 1,2

%A _Altug Alkan_, Feb 06 2016

%E a(8)-a(9) from _Giovanni Resta_, Apr 10 2016

%E a(10)-a(11) from _Chai Wah Wu_, Jul 22 2020