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%I A155562 #22 Jun 25 2024 18:05:30
%S A155562 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,
%T A155562 100,113,121,128,136,137,144,146,153,162,164,169,178,193,194,196,200,
%U A155562 225,226,233,241,242,256,257,272,274,281,288,289,292,306,313,324,328
%N A155562 Intersection of A001481 and A002479: N = a^2 + b^2 = c^2 + 2d^2 for some integers a,b,c,d.
%C A155562 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.
%C A155562 From _Warut Roonguthai_, Oct 13 2009: (Start)
%C A155562 N is also of the form x^2 - 2y^2.
%C A155562 N = (p^2-q^2-2*r*s)^2+(r^2-s^2-2*p*q)^2
%C A155562 = (p^2+q^2-r^2-s^2)^2+2*(p*r-p*s-q*r-q*s)^2
%C A155562 = (p^2+q^2+r^2+s^2)^2-2*(p*r+p*s+q*r-q*s)^2
%C A155562 for some nonnegative integers p, q, r, s. (End)
%C A155562 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
%H A155562 Robert Israel, <a href="/A155562/b155562.txt">Table of n, a(n) for n = 1..10000</a>
%H A155562 Andrew D. Ionaşcu, <a href="https://doi.org/10.2478/auom-2019-0016">Intersecting semi-disks and the synergy of three quadratic forms</a>, An. Şt. Univ. Ovidius Constantą, (2019) Vol. 27, Issue 2, 5-13.
%o A155562 (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}
%o A155562 for( n=1,500, isA155562(n) & print1(n","))
%o A155562 (Python)
%o A155562 from itertools import count, islice
%o A155562 from sympy import factorint
%o A155562 def A155562_gen(): # generator of terms
%o A155562     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))
%o A155562 A155562_list = list(islice(A155562_gen(),30)) # _Chai Wah Wu_, Jun 27 2022
%K A155562 easy,nonn
%O A155562 1,3
%A A155562 _M. F. Hasler_, Jan 24 2009

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