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A269833
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Numbers n such that 2^n + n! is the sum of 2 squares.
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0
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0, 4, 6, 8, 16, 20, 21, 40, 45, 47, 52, 64, 67, 71, 72, 74, 88
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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PROG
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(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))
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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