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%I #62 Jul 12 2023 11:02:35
%S 1,3,9,11,17,19,27,33,41,43,51,57,59,67,73,81,83,89,97,99,107,113,121,
%T 123,129,131,137,139,153,163,171,177,179,187,193,201,209,211,219,227,
%U 233,241,243,249,251,257,267,281,283,289,291,297,307,313,321,323
%N Numbers that are positive integer divisors of 1 + 2*x^2 where x is a positive integer.
%C This sequence is case k=2, A008784 is case k=1, A004613 is case k=4 of divisors of 1 + k*x^2.
%C From Peter M. Chema, May 08 2017 (Start): Also gives the body diagonals of all primitive Pythagorean quadruples that define square prisms, with sides [b, b, and c] and diagonal d, such that 2*b^2 + c^2 = d^2. E.g., sides [2, 2, 1], diagonal 3 = a(2); [4, 4, 7], 9 = a(3); [6, 6, 7], 11 = a(4); [12, 12, 1], 17 = a(5); [6, 6, 17] 19 = a(6); [10, 10, 23], 27 = a(7); [20, 20, 17], 33 = a(8); [24, 24, 23], 41 = a(9)... (a subsequence of A096910) (End)
%C Editorial note: The above comment would be better expressed by talking about right tetrahedra (also called trirectangular tetrahedra), that is, tetrahedra with vertices (b 0 0), (0 c 0), (0 0 d) (here b=c). These are the correct generalizations of Pythagorean triangles. _N. J. A. Sloane_, May 08 2017
%C From _Frank M Jackson_, May 23 2017: (Start)
%C Starting at a(2)=3, this gives the shortest side of a primitive Heronian triangle whose perimeter is 4 times its shortest side. Aka a primitive integer Roberts triangle (see Buchholz link).
%C Also odd and primitive terms generated by x^2 + 2y^2 with x>0 and y>0.
%C Also integers with all prime divisors congruent to 1 or 3 (mod 8). (End)
%H Giovanni Resta, <a href="/A225771/b225771.txt">Table of n, a(n) for n = 1..1000</a>
%H Ralph H. Buchholz, <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.23.2605&rep=rep1&type=pdf">On Triangles with rational altitudes, angle bisectors or medians</a>, Newcastle University (1989), 21-22.
%F a(n) integers whose prime divisors are congruent to 1 or 3 (mod 8). - _Carmine Suriano_, Jan 09 2015; corrected by _Frank M Jackson_, May 23 2017
%t Select[Range[323], False =!= Reduce[1 + 2*x^2 == # y , {x, y}, Integers] &] (* _Giovanni Resta_, Jul 28 2013 *)
%t Select[Range[323], OddQ[#]&&Intersection[{5, 7}, Mod[Divisors[#], 8]]=={} &] (* _Frank M Jackson_, May 23 2017 *)
%o (PARI) {isa(n) = if( n<2, n==1, for( k=1, n\2, if( (1 + 2*k^2)%n == 0, return(1))))} /* _Michael Somos_, Jul 28 2013 */
%Y Cf. A004613, A008784, A096910.
%K nonn
%O 1,2
%A _Michael Somos_, Jul 26 2013
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