A379340 - OEIS

%I #20 Dec 26 2024 20:06:48

%S 0,0,0,0,0,0,1,0,1,1,2,0,5,0,4,4,6,0,11,0,12,7,10,0,23,4,14,10,23,0,

%T 37,0,28,15,24,14,54,0,30,20,57,0,69,0,55,43,44,0,98,12,72,32,76,0,

%U 107,30,107,39,70,0,177,0,80,76,120,40,161,0,128,55,172,0,237

%N a(n) is the number of pairs of similar integer-sided triangles whose perimeters sum up to n.

%C a(n) is the number of solutions to (k + 1)*(x + y + z) = n in positive integers x, y, z and rational k with x <= y <= z and k >= 1.

%C There is no pair of integer-sided similar triangles whose perimeters sum up to a prime. Proof: If the integer perimeters u = x1 + y1 + z1 and v = x2 + y2 + z2 sum up to a prime p, they are coprimes and the scale factor is k = v/u. Since positive integer x1 < u, x2 = x1*v/u is not an integer. It follows that a(p) = 0.

%H Felix Huber, <a href="/A379340/b379340.txt">Table of n, a(n) for n = 0..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SimilarTriangles.html">Similar Triangles</a>

%F a(2*m) = A379341(2*m) + A069905(m), a(2*m-1) = A379341(2*m-1) for nonnegative integers m.

%F a(p) = 0 for prime p.

%e The a(12) = 5 pairs of integer-sided similar triangles whose perimeters sum up to 12 are (1, 1, 1) and (3, 3, 3) with k = 3; (1, 1, 2) and (2, 2, 4) with k = 2; (1, 1, 4) and (1, 1, 4) with k = 1; (1, 2, 3) and (1, 2, 3) with k = 1; (2, 2, 2) and (2, 2, 2) with k = 1.

%e The a(21) = 7 pairs of integer-sided similar triangles whose perimeters sum up to 21 are (1, 1, 1) and (6, 6, 6) with k = 6; (1, 1, 5) and (2, 2, 10) with k = 2; (1, 2, 4) and (2, 4, 8) with k = 2; (1, 3, 3) and (2, 6, 6) with k = 2; (2, 2, 3) and (4, 4, 6) with k = 2; (2, 2, 2) and (5, 5, 5) with k = 5/2; (3, 3, 3) and (4, 4, 4) with k = 4/3.

%p A379340:=proc(n)

%p    local a,p,q,s;

%p    a:=0;

%p    if isprime(n)=false then

%p       for q to n/6 do

%p          for s from 3*q to n/2 by q do

%p             p:=n*q/s-q;

%p             if is(p,integer) and igcd(p,q)=1 then

%p                a:=a+round((s/q)^2/12)

%p             fi	

%p          od

%p       od

%p    fi;

%p    return a

%p end proc;

%p seq(A379340(n),n=0..72);

%Y Cf. A010814, A069905, A070088, A366398, A370599, A379341.

%K nonn

%O 0,11

%A _Felix Huber_, Dec 24 2024