OFFSET
1,3
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.
PROG
(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))
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
M. F. Hasler, Jan 24 2009
STATUS
approved