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A103246 Numbers y, without duplication, in Pythagorean triples x,y,z where x,y,z are relatively prime composite numbers.

%I A103246 #20 Nov 14 2019 19:58:41
%S A103246 21,27,33,55,57,63,75,77,81,87,91,93,99,105,111,115,117,119,123,125,
%T A103246 129,133,135,143,147,153,155,161,165,171,177,183,185,187,189,195,201,
%U A103246 203,207,213,215,217,219,225,235,237,243,247,249,253,255,259,265,267,273
%N A103246 Numbers y, without duplication, in Pythagorean triples x,y,z where x,y,z are relatively prime composite numbers.
%C A103246 The example is the smallest such triple in terms of x. In terms of y, 220^2 + 21^2 = 221^2 is the smallest such triple.
%C A103246 Evidently the triples here are ordered so that x is even and y is odd. - _Robert Israel_, Oct 22 2018
%H A103246 Robert Israel, <a href="/A103246/b103246.txt">Table of n, a(n) for n = 1..10000</a>
%H A103246 MathForFun, <a href="http://groups.yahoo.com/group/mathforfun/message/9962">Pythagorean triples</a>
%H A103246 MathForFun, <a href="/A103246/a103246.pdf">Pythagorean triples</a> [Cached copy]
%H A103246 Chenglong Zou, Peter Otzen, Cino Hilliard, <a href="/A103246/a103246.txt">Pythagorean triplets</a>, digest of 6 messages in mathfun Yahoo group, Mar 19, 2005.
%e A103246 x=16, y=63, 16^2 + 63^2 = 65^2. 63 is the 6th entry in the list.
%p A103246 N:= 1000: # to get all terms <= N
%p A103246 Res:= NULL:
%p A103246 for m from 1 to N by 2 do
%p A103246   for n from 1 to m-2 by 2 while m*n <= N do
%p A103246     if igcd(m,n) > 1 then next fi;
%p A103246     if not isprime(m*n) and not isprime((m^2+n^2)/2) then
%p A103246       Res:= Res, m*n;
%p A103246     fi
%p A103246 od od:
%p A103246 sort(convert({Res},list)); # _Robert Israel_, Oct 22 2018
%o A103246 (PARI) pythtri(n) = { local(a,b,c=0,k,x,y,z,vy,j); w = vector(n*n); for(a=1,n, for(b=1,n, x=2*a*b; y=b^2-a^2; z=b^2+a^2; if(y > 0 &!isprime(x) &!isprime(y) &!isprime(z), if(gcd(x,y)==1&gcd(x,z)==1&gcd(y,z)==1, c++; w[c]=y; ) ) ) ); vy=vector(c); w=vecsort(w); for(j=1,n*n, if(w[j]>0, k++; vy[k]=w[j]; ) ); for(j=1,200, if(vy[j+1]<>vy[j],print1(vy[j]",")) ) }
%K A103246 easy,nonn
%O A103246 1,1
%A A103246 _Cino Hilliard_, Mar 19 2005