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%I #38 Nov 05 2021 09:37:12
%S 52,700,740,996,3364,6240,7800,8400,10952,11184,11352,11492,11484,
%T 13156,20280,20988,21320,22472,26180,26588,28168,34500,39988,40680,
%U 43700,44944,45976,49500,58956,70448,77500,90168,103896,105468,106200,115752,118636,124620,129000
%N Primitive integer length of the side of an origin-centered square that contains inside its boundary a point with integer coordinates that is an integer distance from three of the four corners.
%C No point with integer distance to all four corners is known.
%C The sequence only contains even values because an odd-sided square centered at the origin has corners with non-integer coordinates, which cannot be at integer distance from interior lattice points. If the square instead of being centered at the origin has a corner on the origin, then the resulting sequence is A260549. - _Giovanni Resta_, Jul 29 2015
%H Yasushi Ieno, <a href="https://arxiv.org/abs/2111.02888">Other special cases of a square problem</a>, arXiv:2111.02888 [math.GM], 2021.
%H Yang Ji, <a href="https://arxiv.org/abs/2105.05250">Several special cases of a square problem</a>, arXiv:2105.05250 [math.GM], 2021.
%H G. Shute and K. L. Yocom, <a href="https://www.jstor.org/stable/2689508">Problem 966. Seven integral distances</a>, Math. Mag. 50, 166 (1977).
%H UnsolvedProblems Web Site, <a href="http://unsolvedproblems.org/index_files/RationalDistance.htm">Rational Distance</a>
%e With n = side length, we find an a,b such that a^2 + b^2 = d1^2, a^2 + (n-b)^2 = d2^2, b^2 + (n-a)^2 = d3^2, (n-a)^2 + (n-b)^2 = d4^2 is true in integers for three of these four equations. n = 52 is the first, with a=7 and b=24.
%o (PARI) has(n)=for(a=1,n-1,for(b=a,n-1, if(issquare(norml2([a,b])) + issquare(norml2([n-a,b])) + issquare(norml2([a,n-b])) + issquare(norml2([n-a,n-b])) > 2, return(1)))); 0
%o is(n)=sumdiv(n,d,has(d))==1 \\ _Charles R Greathouse IV_, Jul 28 2015
%Y Cf. A260549.
%K nonn
%O 1,1
%A _Mark Underwood_, Aug 08 2012
%E Data corrected and name improved by _Mark Underwood_, Jul 28 2015
%E a(7)-a(39) from _Giovanni Resta_, Jul 29 2015
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