OFFSET
1,2
COMMENTS
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)
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
From Frank M Jackson, May 23 2017: (Start)
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).
Also odd and primitive terms generated by x^2 + 2y^2 with x>0 and y>0.
Also integers with all prime divisors congruent to 1 or 3 (mod 8). (End)
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..1000
Ralph H. Buchholz, On Triangles with rational altitudes, angle bisectors or medians, Newcastle University (1989), 21-22.
FORMULA
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
MATHEMATICA
Select[Range[323], False =!= Reduce[1 + 2*x^2 == # y , {x, y}, Integers] &] (* Giovanni Resta, Jul 28 2013 *)
Select[Range[323], OddQ[#]&&Intersection[{5, 7}, Mod[Divisors[#], 8]]=={} &] (* Frank M Jackson, May 23 2017 *)
PROG
(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 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Jul 26 2013
STATUS
approved