

A225771


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


1



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
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OFFSET

1,2


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
Comment 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), 2122.


FORMULA

a(n) integers whose prime divisors are congruent to 1 or 3 (mod 8).  Carmine Suriano, Jan 09 2015


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

Cf. A004613, A008784, A096910.
Sequence in context: A190238 A047471 A201544 * A215816 A211864 A157139
Adjacent sequences: A225768 A225769 A225770 * A225772 A225773 A225774


KEYWORD

nonn


AUTHOR

Michael Somos, Jul 26 2013


EXTENSIONS

Formula corrected by Frank M Jackson, May 23 2017


STATUS

approved



