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%I #14 Nov 16 2019 13:59:20
%S 3,6,9,12,18,24,27,36,48,54,56,60,72,81,96,108,112,120,144,162,180,
%T 192,216,224,240,243,288,300,319,324,360,384,392,399,432,448,480,486,
%U 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
%N 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.
%C 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.
%H Cino Hilliard and others, <a href="/A156997/a156997.txt">Pythagorean Theorem and prime factors</a>, digest of 9 messages in mathforfun Yahoo group, Feb 20 - Feb 21, 2009. [Cached copy]
%H MathForFun, <a href="http://groups.yahoo.com/group/mathforfun/message/13635">Pythagorean triple digital sums</a>.
%F Let a,b,c, m > u,k be positive integers. If a^2 + b^2 = c^2 then
%F a = k*2mu,b=k*(m^2-u^2) and c=k*(m^2+u^2).
%e 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.
%e So 319 is in the sequence.
%o (PARI) pythsum(n) =
%o {
%o local(v,x,a,b,cm,k,u,s,ct=0);
%o v=vector(400);
%o for(m=1,n+n,
%o for(u=1,n,
%o for(k=1,n,
%o a=2*m*u*k;
%o if(u>m,b=k*(u^2-m^2),b=k*(m^2-u^2));
%o c=k*(m^2+u^2);
%o fa=ifactord(a);
%o fb=ifactord(b);
%o fc=ifactord(c);
%o s=0;
%o s2=0;
%o for(a1=1, length(fa), s+=fa[a1]);
%o for(b1=1, length(fb), s+=fb[b1]);
%o for(c1=1, length(fc), s2+=fc[c1]);
%o if(s==s2&&b>0,
%o \\print(m","u","k","a", "b", "c", " fa" + "fb" = "fc);
%o ct++;
%o if(a<b,v[ct]=a,v[ct]=b)
%o )
%o )
%o )
%o );
%o y=vecsort(v);
%o for(x=1,399,if(y[x]>0&&y[x]<>y[x+1],print1(y[x]",")));
%o }
%o ifactord(n) = /* The vector of the distinct integer factors of n. */
%o {
%o local(f,j,k,flist);
%o flist=[];
%o f=Vec(factor(n));
%o for(j=1,length(f[1]),
%o flist = concat(flist,f[1][j])
%o );
%o return(flist)
%o }
%K nonn
%O 1,1
%A _Cino Hilliard_, Feb 20 2009
%E Based on suggestions from _Robert G. Wilson v_, corrected sequence, definition, comments, formula and Pari program. - _Cino Hilliard_, Apr 03 2009
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