A108707 - OEIS

%I #21 Apr 10 2021 22:29:29

%S 0,0,0,0,3,0,0,0,0,6,0,0,5,0,9,0,8,0,0,12,0,0,0,0,7,10,0,0,20,18,0,0,

%T 0,16,21,0,12,0,15,24,9,0,0,0,27,0,0,0,0,14,24,20,28,0,33,0,0,40,0,36,

%U 11,0,0,0,16,0,0,32,0,42,0,0,48,24,21,0,0,30,0,48,0,18,0,0,13,0,60,0,39,54

%N Minimum side in Pythagorean triangles with hypotenuse of n.

%H David A. Corneth, <a href="/A108707/b108707.txt">Table of n, a(n) for n = 1..10000</a>

%e a(5) = 3 as the right triangle with sides (3, 4, 5) has hypotenuse n = 5 smallest side a(5) = 3. This is the smallest side a right triangle with integer sides and hypotenuse 5 can have. - _David A. Corneth_, Apr 10 2021

%t f[n_]:=Block[{k=n-1,m=Sqrt[n/2],a},While[k>m&&!IntegerQ[(a=Sqrt[n^2-k^2])],k--];If[k<=m,0,a]];Table[f[n],{n,90}]

%o (PARI) first(n) = {my(lh = List(), res = vector(n, i, oo)); for(u = 2, sqrtint(n), for(v = 1, u, if (u^2+v^2 > n, break); if ((gcd(u, v) == 1) && (0 != (u-v)%2), for (i = 1, n, if (i*(u^2+v^2) > n, break); listput(lh, i*(u^2+v^2)); res[i*(u^2+v^2)] = vecmin([res[i*(u^2+v^2)], i*(u^2 - v^2), i*2*u*v]))))); for(i = 1, n, if(res[i] == oo, res[i] = 0)); res } \\ _David A. Corneth_, Apr 10 2021, adapted from A009000

%Y Cf. A083025, A046083, A046084, A009000, A009003, A108708.

%Y A046080 gives the number of Pythagorean triangles with hypotenuse n.

%K nonn

%O 1,5

%A _Sébastien Dumortier_, Jun 20 2005

%E Extended by _Ray Chandler_, Dec 20 2011