A376900 -id:A376900 - cp's OEIS frontend

A379595 Numbers k for which A376900(k) = k.

0, 385, 386, 387, 390, 392, 404, 405, 406

Offset: 1

Felix Huber, Dec 26 2024

The primitive Pythagorean triples cause A376900(n) to grow slightly overproportionally in relation to n, apart from minor fluctuations. This is why this sequence is finite and full.

Felix Huber, Right Triangles belonging to A376900(k)=k

Cf. A376900.

A379595:=proc(k)
   local a,p,q,v,m;
   a:=0;
   for p from 2 to evalf(sqrt(sqrt(2)*k+1)) do
      for q from 1 to min(p-1,floor(k/(sqrt(2)*p))) do
         if gcd(p,q)=1 and is(p+q,odd) then
            v:=max(p^2-q^2,2*p*q);
            m:=min(p^2-q^2,2*p*q)/v;
            a:=a+floor(k/v*sqrt(m^2-2*m+2));
         fi
      od
   od;
   if a=k then
      return k
   fi
end proc;
seq(A379595(k),k=0..1000);

A379663 a(n) is the number of integer-sided triangles whose sides are in geometric progression with smallest side n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 2, 4, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 5, 4, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 5, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 4, 2, 1, 1, 1, 3, 6, 1, 1, 2, 1, 1, 1, 2

Offset: 1

Felix Huber, Jan 07 2025

nonn

The integer sides of the triangles are n, n*r, n*r^2 with rational r >= 1. From the triangle inequality n + n*r >= n*r^2 follows r <= (1 + sqrt(5))/2 (golden ratio). Therefore 1 <= r = c/d < (1 + sqrt(5))/2, where c and d are coprimes and d^2 divides n.

			The a(18) = 2 integer-sided triangles whose sides form a geometric sequence are [18, 18, 18] with r = 1, [18, 24, 32] with r = 4/3.
The a(25) = 4 integer-sided triangles whose sides form a geometric sequence are [25, 25, 25] with r = 1, [25, 30, 36] with r = 6/5, [25, 35, 49] with r = 7/5, [25, 40, 64] with r = 8/5.
The a(36) = 4 integer-sided triangles whose sides form a geometric sequence are [36, 36, 36] with r = 1, [36, 54, 81] with r = 3/2, [36, 48, 64] with r = 4/3, [36, 42, 49] with r = 7/6.
See also the linked Maple program "Triangles for a given n".

Felix Huber, Table of n, a(n) for n = 1..10000
Felix Huber, Triangles for a given n
Eric Weisstein's World of Mathematics, Golden Ratio
Wikipedia, Geometric Progression
Wikipedia, Triangle Inequality

Cf. A000188, A000201, A008833, A027750, A046951, A057918, A060143, A366398, A370599, A376900, A379033, A379340, A379341, A379705.

A379663:=n->floor(2*expand(NumberTheory:-LargestNthPower(n,2))/(1+sqrt(5)))+1;
seq(A379663(n),n=1..88);

a(n) = A060143(A000188(n)) + 1.

cp's OEIS Frontend

A379595 Numbers k for which A376900(k) = k.

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