A106430 Ordered and uniqued length of side opposite the greater of the two angles, one being the double of the other, of a primitive integer-sided triangle.
6, 12, 15, 20, 28, 30, 35, 40, 42, 45, 56, 63, 66, 70, 72, 77, 84, 88, 90, 91, 99, 104, 110, 117, 120, 126, 130, 132, 143, 144, 153, 154, 156, 165, 170, 176, 182, 187, 190, 195, 198, 204, 208, 209, 210, 220, 221, 228, 231, 234, 238, 240, 247, 255, 260, 266, 272, 273, 276, 285, 286, 299
Offset: 1
Keywords
Links
- Albert Lau, Table of n, a(n) for n = 1..10539
Programs
-
Mathematica
aMax = 300; Select[Sequence @@@ Table[{m (m + n), n (m + n), m n}, {m, Sqrt[aMax]}, {n, Min[m - 1, aMax/m - m]}], GCD @@ # == 1 &] // Sort; %[[;; , 1]] (* Albert Lau, Sep 19 2014 *)
Formula
Values r*s, where r
The other 2 sides are s^2 and r^2-s^2. - Albert Lau, Sep 19 2014
A246429 List of c, where c is positive integer solutions of 1/a + 1/b = 1/c and a>b>c and gcd(a,b,c)=1, sorted by ascending a, then b.
2, 3, 6, 4, 12, 5, 10, 15, 6, 20, 7, 14, 30, 21, 8, 28, 35, 24, 9, 42, 18, 40, 10, 36, 56, 45, 30, 11, 22, 63, 72, 33, 12, 44, 70, 55, 13, 66, 90, 26, 77, 60, 39, 88, 14, 99, 52, 84, 110, 65, 42, 15, 78, 30, 91, 70, 16, 104, 132, 60, 117, 130
Offset: 1
Comments
Examples
1/6 + 1/3 = 1/2. 1/12 + 1/4 = 1/3. 1/15 + 1/10 = 1/6.
Links
- Ivan Neretin, Table of n, a(n) for n = 1..10000
Programs
-
Mathematica
aMax = 300; Select[Sequence @@@ Table[{m (m + n), n (m + n), m n}, {m, Sqrt[aMax]}, {n, Min[m - 1, aMax/m - m]}], GCD @@ # == 1 &] // Sort; %[[;; , 3]]
A280680 The square roots of the radii (a, b, c) of three primitive mutually tangent circles all touching the same straight line, with a > b > c.
6, 3, 2, 12, 4, 3, 15, 10, 6, 20, 5, 4, 28, 21, 12, 30, 6, 5, 35, 14, 10, 40, 24, 15, 42, 7, 6, 45, 36, 20, 56, 8, 7, 63, 18, 14, 66, 55, 30, 70, 30, 21, 72, 9, 8, 77, 44, 28, 84, 60, 35, 88, 33, 24, 90, 10, 9, 91, 78, 42, 99, 22, 18, 104, 65, 40
Offset: 1
Examples
The first few triples are [6, 3, 2], [12, 4, 3], [15, 10, 6], [20, 5, 4].
Links
- Colin Barker, Table of n, a(n) for n = 1..3000
- Eric Weisstein's World of Mathematics, Tangent Circles
Programs
-
PARI
a280680(amax) = { my(L=List()); for(a=1, amax, for(b=1, a-1, c=(1/(1/a + 1/b))^2; if(type(c)=="t_INT" && gcd([a^2,b^2,c])==1, listput(L, [a,b,sqrtint(c)]) ) ) ); Vec(L) } concat(a280680(100))
Comments