A155562 Intersection of A001481 and A002479: N = a^2 + b^2 = c^2 + 2d^2 for some integers a,b,c,d.

0, 1, 2, 4, 8, 9, 16, 17, 18, 25, 32, 34, 36, 41, 49, 50, 64, 68, 72, 73, 81, 82, 89, 97, 98, 100, 113, 121, 128, 136, 137, 144, 146, 153, 162, 164, 169, 178, 193, 194, 196, 200, 225, 226, 233, 241, 242, 256, 257, 272, 274, 281, 288, 289, 292, 306, 313, 324, 328

Offset: 1

Author

M. F. Hasler, Jan 24 2009

Keywords

• easy
• nonn

Comments

Contains A155561 as a subsequence (obtained by restricting a,b,c,d to be nonzero). Also contains A000290 (squares) and A001105 (twice the squares) as subsequence.

From Warut Roonguthai, Oct 13 2009: (Start)

N is also of the form x^2 - 2y^2.

N = (p^2-q^2-2*r*s)^2+(r^2-s^2-2*p*q)^2

= (p^2+q^2-r^2-s^2)^2+2*(p*r-p*s-q*r-q*s)^2

= (p^2+q^2+r^2+s^2)^2-2*(p*r+p*s+q*r-q*s)^2

for some nonnegative integers p, q, r, s. (End)

Numbers k such that in the prime factorization of k, all odd primes that occur with an odd exponent are congruent to 1 (mod 8). - Robert Israel, Jun 24 2024

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Andrew D. Ionaşcu, Intersecting semi-disks and the synergy of three quadratic forms, An. Şt. Univ. Ovidius Constantą, (2019) Vol. 27, Issue 2, 5-13.

Programs

PARI

isA155562(n,/* use optional 2nd arg to get other analogous sequences */c=[2,1]) = { for(i=1,#c, for(b=0,sqrtint(n\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=1,500, isA155562(n) & print1(n","))

Python

from itertools import count, islice
from sympy import factorint
def A155562_gen(): # generator of terms
    return filter(lambda n:all((p & 3 != 3 and p & 7 < 5) or e & 1 == 0 for p, e in factorint(n).items()),count(0))
A155562_list = list(islice(A155562_gen(),30)) # Chai Wah Wu, Jun 27 2022