A201223 Primitive Eisenstein triples (a,b,c) listed as groups of three in order of increasing b.

3, 8, 7, 5, 8, 7, 7, 15, 13, 8, 15, 13, 5, 21, 19, 16, 21, 19, 11, 35, 31, 24, 35, 31, 7, 40, 37, 33, 40, 37, 13, 48, 43, 35, 48, 43, 16, 55, 49, 39, 55, 49, 9, 65, 61, 56, 65, 61, 32, 77, 67, 45, 77, 67, 17, 80, 73, 63, 80, 73, 40, 91, 79, 51, 91, 79, 11, 96, 91

Offset

1,1

Comments

An Eisenstein triple is a triple (a,b,c) of positive integers with a<c<b and a^2 - a*b + b^2 = c^2. It is primitive if a, b, and c are relatively prime.

Links

Danny Rorabaugh, Table of n, a(n) for n = 1..1494

Russell A. Gordon, Properties of Eisenstein Triples, Mathematics Magazine 85 (2012), 12-25.

Example

(a,b,c)=(3,8,7) is an Eisenstein triple since 3<7<8 and 3^2 - 3*8 + 8^2 = 7^2.  GCD(3,8,7) = 1, so the triple is primitive.  No Eisenstein triple exists with b<8, so a(1)=3, a(2)=8, a(3)=7.

Mathematica

x = {}; For[b = 1, b <= 77, b++, For[c = 1, c < b, c++, For[a = 1, a < c, a++, {If[(a^2 - a*b + b^2 == c^2) && (GCD[a, b, c] == 1), AppendTo[x, {a, b, c}]]}]]]; Flatten[x]

Crossrefs

Cf. A121992.

Sequence in context: A335810 A225016 A121992 * A195721 A021262 A276120

Adjacent sequences: A201220 A201221 A201222 * A201224 A201225 A201226

Keyword

nonn

Author

John W. Layman, May 09 2012

Status

approved