A358311 Lucas numbers that are not the sum of two squares.

3, 7, 11, 47, 76, 123, 199, 322, 843, 1364, 2207, 3571, 5778, 15127, 24476, 39603, 64079, 103682, 167761, 271443, 439204, 710647, 1149851, 4870847, 7881196, 12752043, 20633239, 33385282, 87403803, 141422324, 228826127, 370248451, 599074578, 1568397607, 2537720636

Offset: 1

Author

Chai Wah Wu, Jan 10 2023

Keywords

• nonn

Comments

Lucas numbers with indices 2, 4, 5 mod 6 are 3 mod 4, so these are all terms. - Charles R Greathouse IV, Jan 11 2023

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..958

Crossrefs

Intersection of A000032 and A022544.

Cf. A356809.

Programs

Maple

R:= NULL: count:= 0:
a:= 2: b:= 1:
for i from 1 while count < 100 do
  a, b:= b,a+b;
    if ormap(t -> t[2]::odd and t[1] mod 4 = 3, ifactors(b)[2]) then
     R:= R, b; count:= count+1
  fi
od:
R; # Robert Israel, Jan 10 2023

Python

from sympy import factorint
from itertools import islice
def A358311_gen(): # generator of terms
    a, b = 2,1
    while True:
        if any(e&1 and p&3==3 for p, e in factorint(a).items()):
            yield a
        a, b = b, a+b
A358311_list = list(islice(A358311_gen(),40))

Formula

phi^n < a(n) < phi^(2n) for n > 4. - Charles R Greathouse IV, Jan 11 2023