A104461 Number of instances of nonprimes m in Pythagorean triples x,y,z such that x^2 + y^2 = z^2. Except for 1, the number of instances of composite numbers m in Pythagorean triples.

0, 1, 1, 2, 2, 2, 4, 1, 5, 3, 2, 5, 4, 1, 7, 4, 2, 3, 4, 5, 4, 4, 2, 5, 7, 1, 5, 8, 4, 4, 8, 1, 10, 2, 4, 5, 5, 3, 5, 7, 4, 2, 14, 1, 7, 5, 8, 4, 5, 4, 5, 12, 2, 9, 4, 4, 5, 11, 4, 2, 13, 8, 1, 5, 7, 8, 5, 4, 4, 1, 5, 13, 2, 7, 9, 5, 8, 14, 2, 10, 5, 5, 10, 4, 5, 5, 8, 1, 5, 23, 2, 2, 5, 4, 6, 7, 6, 4, 8, 13

Offset

1,4

Comments

The PARI script is direct and very fast for m = x,y values but slows in the trial routine for m = z. We save some time for m even allowing the testing of only even values of y.

Links

Table of n, a(n) for n=1..100.

Formula

Consider Pythagorean triples x^2 + y^2 = z^2. We seek to find the total number of instances of an integer m being x or y or z. The solution for x or y is straightforward by considering appropriate lesser and greater pairwise factors, L, G of m^2 in z^2 - y^2 = (z-y)(z+y) = m^2. Then solve for z and y with the relations, z-y = L z+y = G 2z = L+G, z = (L+G)/2 where L and G are both even if m is even or both odd if m is odd. The number of L factors < m is the number of instances of x or y. The count of instances z=m is solved by trial on x^2 = m^2 - y^2.

Example

For m=30 there are 5 Pythagorean triples that have a 30:
  30, 224, 226
  30,  72,  78
  30,  40,  50
  30,  16,  34
  18,  24,  30

Prog

(PARI)
\\ instances of m in Pythagorean triples using a direct method for x, y
pythm3(m) = { local(m2, ln, j, j2=0, d, d2, q2, q, a, b, x, x1, x2, xx, y, y2, z, c, c2, r, f, str, stp); d=divisors(m^2); /* get the divisors of m^2 */ ln=length(d)-1; d2=q2=vector(ln); m2=m^2; if(m%2, r=1, r=0); for(j=1, ln, /* save only the both even r=0, both odd r=1 */ if(d[j]%2==r, if(m2/d[j]%2==r, j2++; d2[j2]=d[j]; q2[j2]=m2/d[j]; /* save m/factor to solve (z-y)(z+y) = m^2 */ ) ) ); x2=y2 = vector(20); for(j=1, j2, z=(d2[j] + q2[j])/2; y= z - d2[j]; if(y>0, c++; ) ); if(m%2==0, start=2; step=2, start=1; step=1); forstep(y=start, m-1, step, /* esolve when z is m */ x1 = (m2-y^2); if(issquare(x1), c2++; x2[c2]=floor(sqrt(x1)); /* save to later mask dupes */ y2[c2]=y; ) ); for(x=1, c2, /* mask the dupes routine */ for(y=x, c2, if(x2[x]==y2[y], ) ) ); return(c+c2/2) /* print total */}
for(k=1, 400, if(isprime(k)==0, print1(pythm3(k)", ")))

Crossrefs

Cf. A046081, A088978.

Sequence in context: A358318 A071470 A197116 * A308042 A335694 A084862

Adjacent sequences: A104458 A104459 A104460 * A104462 A104463 A104464

Keyword

easy,nonn

Author

Cino Hilliard, Apr 18 2005

Status

approved