A269833 Numbers n such that 2^n + n! is the sum of 2 squares.

0, 4, 6, 8, 16, 20, 21, 40, 45, 47, 52, 64, 67, 71, 72, 74, 88

Offset

1,2

Comments

Integers n such that the equation 2^n + n! = x^2 + y^2 where x and y are integers is solvable.

4, 8, 16 and 64 are powers of 2. What is the next power of 2 (if any) in this sequence?

103 <= a(18) <= 108. 108, 117, 144, 176, 254, 537 are terms. - Chai Wah Wu, Jul 22 2020

Links

Table of n, a(n) for n=1..17.

Example

6 is a term because 2^6 + 6! = 28^2.
8 is a term because 2^8 + 8! = 24^2 + 200^2.
21 is a term because 2^21 + 21! = 1222129664^2 + 7042537984^2.

Mathematica

Select[Range[0, 64], SquaresR[2, 2^# + #!] > 0 &] (* Michael De Vlieger, Mar 07 2016 *)

Prog

(PARI) isA001481(n) = #bnfisintnorm(bnfinit(z^2+1), n);
for(n=0, 1e2, if(isA001481(n!+2^n), print1(n, ", ")));
(Python)
from math import factorial
from itertools import count, islice
from sympy import factorint
def A269833_gen(): # generator of terms
    return filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in factorint((1<<n)+factorial(n)).items()), count(0))
A269833_list = list(islice(A269833_gen(), 9)) # Chai Wah Wu, Jun 27 2022

Crossrefs

Cf. A001481, A007611.

Sequence in context: A055397 A239412 A295006 * A049421 A260314 A238269

Adjacent sequences: A269830 A269831 A269832 * A269834 A269835 A269836

Keyword

nonn,more

Author

Altug Alkan, Mar 06 2016

Extensions

a(17) from Chai Wah Wu, Jul 22 2020

Status

approved