A156997 Smaller of a and b if a^2+b^2=c^2 and the sum of the distinct prime divisors of a and b is the sum of the distinct prime divisors of c.

3, 6, 9, 12, 18, 24, 27, 36, 48, 54, 56, 60, 72, 81, 96, 108, 112, 120, 144, 162, 180, 192, 216, 224, 240, 243, 288, 300, 319, 324, 360, 384, 392, 399, 432, 448, 480, 486, 504, 540, 576, 600, 638, 648, 720, 728, 729, 768, 784, 798, 864, 896, 900, 957, 972, 1008, 1152, 1176, 1276, 1296, 1400, 1456, 1458

Offset

1,1

Comments

15 divides abc. Since we define a=2mu and b=m^2-b^2, the order of the triple that generates a(n) can have the first term greater than the second term. For this reason I chose to list the smaller of a and b. The idea for this sequence comes from a post in the Yahoo group mathforfun which is in the link.

Links

Table of n, a(n) for n=1..63.

Cino Hilliard and others, Pythagorean Theorem and prime factors, digest of 9 messages in mathforfun Yahoo group, Feb 20 - Feb 21, 2009. [Cached copy]

MathForFun, Pythagorean triple digital sums.

Formula

Let a,b,c, m > u,k be positive integers. If a^2 + b^2 = c^2 then

a = k*2mu,b=k*(m^2-u^2) and c=k*(m^2+u^2).

Example

360^2+319^2=481^2. 360=2^3*3^2*5,319=11*29 and 481=13*37. 2+3+5+11+29=13+37.
So 319 is in the sequence.

Prog

(PARI) pythsum(n) =
{
local(v, x, a, b, cm, k, u, s, ct=0);
v=vector(400);
for(m=1, n+n,
for(u=1, n,
for(k=1, n,
a=2*m*u*k;
if(u>m, b=k*(u^2-m^2), b=k*(m^2-u^2));
c=k*(m^2+u^2);
fa=ifactord(a);
fb=ifactord(b);
fc=ifactord(c);
s=0;
s2=0;
for(a1=1, length(fa), s+=fa[a1]);
for(b1=1, length(fb), s+=fb[b1]);
for(c1=1, length(fc), s2+=fc[c1]);
if(s==s2&&b>0,
\\print(m", "u", "k", "a", "b", "c", " fa" + "fb" = "fc);
ct++;
if(a<b, v[ct]=a, v[ct]=b)
)
)
)
);
y=vecsort(v);
for(x=1, 399, if(y[x]>0&&y[x]<>y[x+1], print1(y[x]", ")));
}
ifactord(n) = /* The vector of the distinct integer factors of n. */
{
local(f, j, k, flist);
flist=[];
f=Vec(factor(n));
for(j=1, length(f[1]),
flist = concat(flist, f[1][j])
);
return(flist)
}

Crossrefs

Sequence in context: A310156 A231960 A052287 * A063996 A065119 A293396

Adjacent sequences: A156994 A156995 A156996 * A156998 A156999 A157000

Keyword

nonn

Author

Cino Hilliard, Feb 20 2009

Extensions

Based on suggestions from Robert G. Wilson v, corrected sequence, definition, comments, formula and Pari program. - Cino Hilliard, Apr 03 2009

Status

approved