A225771 Numbers that are positive integer divisors of 1 + 2*x^2 where x is a positive integer.

1, 3, 9, 11, 17, 19, 27, 33, 41, 43, 51, 57, 59, 67, 73, 81, 83, 89, 97, 99, 107, 113, 121, 123, 129, 131, 137, 139, 153, 163, 171, 177, 179, 187, 193, 201, 209, 211, 219, 227, 233, 241, 243, 249, 251, 257, 267, 281, 283, 289, 291, 297, 307, 313, 321, 323

Offset: 1

Author

Michael Somos, Jul 26 2013

Keywords

• nonn

Comments

This sequence is case k=2, A008784 is case k=1, A004613 is case k=4 of divisors of 1 + k*x^2.

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.

Crossrefs

Cf. A004613, A008784, A096910.

Programs

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 *)

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 */

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