|
| |
|
|
A057102
|
|
Congrua (possible solutions to the congruum problem): numbers n such that there are integers x, y and z with n = x^2-y^2 = z^2-x^2.
|
|
13
| |
|
|
24, 96, 120, 240, 336, 384, 480, 720, 840, 960, 1320, 1344, 1536, 1920, 1944, 2016, 2184, 2520, 2880, 3360, 3696, 3840, 3960, 4896, 5280, 5376, 5544, 6144, 6240, 6840, 6864, 7680, 7776, 8064, 8736, 9240, 9360, 9720, 10080, 10296, 10920, 11520, 12144
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,1
|
|
|
COMMENTS
| Each congruum is a multiple of 24; it cannot be a square.
Numbers of the form (4(x^3y-xy^3) (where x,y are integrs and x>=y). Squares of these numbers are of the form N^4-K^2 (where N belongs to A135786 and K to A135789 or A135790). Proof uses identity: (4(x^3y-xy^3))^2=(x^2+y^2)^4-(x^4 - 6x^2 y^2 + y^4)^2 - Artur Jasinski (grafix(AT)csl.pl), Nov 29 2007, Nov 14 2008
|
|
|
LINKS
| Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
|
|
|
FORMULA
| a(n) = 4 * A073120(n) [From Max Alekseyev (maxale(AT)gmail.com), Nov 14 2008]
|
|
|
EXAMPLE
| a(9)=840 since 840=29^2-1^2=41^2-29^2 (indeed also 840=37^2-23^2=47^2-37^2)
|
|
|
MATHEMATICA
| a = {}; Do[Do[w = 4x^3y - 4x y^3; If[w > 0 && w < 10000, AppendTo[a, w]], {x, y, 1000}], {y, 1, 1000}]; Union[a] - Artur Jasinski (grafix(AT)csl.pl), Nov 29 2007
|
|
|
CROSSREFS
| Cf. A004431 for possible values of x in definition. Cf. A057103, A055096 for triangles of all congrua and values of x.
Cf. A073120, A135789, A135786.
Sequence in context: A090214 A103251 A198387 * A057103 A055669 A195824
Adjacent sequences: A057099 A057100 A057101 * A057103 A057104 A057105
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Henry Bottomley (se16(AT)btinternet.com), Aug 02 2000
|
| |
|
|
