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A378675 Areas of trapezoids with exactly one pair of parallel sides having prime sides and height.

%I A378675 #20 Dec 20 2024 13:20:04
%S A378675 15,21,27,27,45,45,55,63,65,81,85,85,95,99,115,117,125,125,135,145,
%T A378675 155,171,175,175,185,189,205,207,225,235,243,245,265,275,279,295,297,
%U A378675 315,315,325,333,335,355,365,385,387,405,407,425,451,455,459,473,475,475
%N A378675 Areas of trapezoids with exactly one pair of parallel sides having prime sides and height.
%H A378675 Felix Huber, <a href="/A378675/b378675.txt">Table of n, a(n) for n = 1..2633</a>
%H A378675 Felix Huber, <a href="/A378675/a378675.txt">Trapezoids having prime sides and height with area A</a>
%e A378675 27 is twice in the sequence because there are two distinct trapezoids [p, d, q, f, h] (p and q are parallel, height h) with prime sides and height and area 27: [13, 5, 5, 5, 3], [11, 3, 7, 5, 3].
%p A378675 with(NumberTheory):
%p A378675 A378675:=proc(A)
%p A378675    local m,p,q,i,j,d,f,h,x,y,M,T;
%p A378675    if isprime(A)=false and A>1 then
%p A378675       T:=[];
%p A378675       M:=map(x->A/x,select(isprime,(Divisors(A)) minus {2}));
%p A378675       for m in M do
%p A378675          for i to pi(floor(m-1/2)) do
%p A378675             q:=ithprime(i);
%p A378675             p:=2*m-q;
%p A378675             if isprime(p) then
%p A378675                h:=A/m;
%p A378675 	       for x from max(4,floor((p-q+1)/2)) by 2 to (h^2-1)/2 do
%p A378675 	          y:=p-q-x;
%p A378675 	          if issqr(x^2+h^2) and issqr(y^2+h^2) then
%p A378675 	             d:=isqrt(y^2+h^2);
%p A378675 	             f:=isqrt(x^2+h^2);
%p A378675 	             if isprime(d) and isprime(f) then
%p A378675 	                T:=[op(T),A]
%p A378675 	             fi
%p A378675 	          fi
%p A378675 	       od
%p A378675 	    fi
%p A378675          od
%p A378675       od;
%p A378675       return op(T)
%p A378675    fi;
%p A378675 end proc;
%p A378675 seq(A378675(A),A=1..475);
%Y A378675 Cf. A000040, A027750, A070088, A214602, A365049, A366398, A374594, A378148, A378149, A378150.
%K A378675 nonn
%O A378675 1,1
%A A378675 _Felix Huber_, Dec 04 2024