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

15, 21, 27, 27, 45, 45, 55, 63, 65, 81, 85, 85, 95, 99, 115, 117, 125, 125, 135, 145, 155, 171, 175, 175, 185, 189, 205, 207, 225, 235, 243, 245, 265, 275, 279, 295, 297, 315, 315, 325, 333, 335, 355, 365, 385, 387, 405, 407, 425, 451, 455, 459, 473, 475, 475

Offset: 1

Author

Felix Huber, Dec 04 2024

Keywords

• nonn

Examples

			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].
		

Links

• Felix Huber, Table of n, a(n) for n = 1..2633
• Felix Huber, Trapezoids having prime sides and height with area A

Crossrefs

Cf. A000040, A027750, A070088, A214602, A365049, A366398, A374594, A378148, A378149, A378150.

Programs

Maple

with(NumberTheory):
A378675:=proc(A)
   local m,p,q,i,j,d,f,h,x,y,M,T;
   if isprime(A)=false and A>1 then
      T:=[];
      M:=map(x->A/x,select(isprime,(Divisors(A)) minus {2}));
      for m in M do
         for i to pi(floor(m-1/2)) do
            q:=ithprime(i);
            p:=2*m-q;
            if isprime(p) then
               h:=A/m;
	       for x from max(4,floor((p-q+1)/2)) by 2 to (h^2-1)/2 do
	          y:=p-q-x;
	          if issqr(x^2+h^2) and issqr(y^2+h^2) then
	             d:=isqrt(y^2+h^2);
	             f:=isqrt(x^2+h^2);
	             if isprime(d) and isprime(f) then
	                T:=[op(T),A]
	             fi
	          fi
	       od
	    fi
         od
      od;
      return op(T)
   fi;
end proc;
seq(A378675(A),A=1..475);