A385976 - OEIS

%I #46 Aug 28 2025 22:02:14

%S 12,70,98,2002,2842,20026,91698,721786

%N Least integer k such that there are exactly n primitive Heron triangles having the same area and perimeter k.

%H BBS Math Blog, <a href="https://bbs.emath.ac.cn/forum.php?mod=viewthread&amp;tid=17031">Primitive Heron triangles with equal perimeter and area</a> (in Chinese).

%e a(3) = 98 because there exists 3 primitive Heron triangles: {{29, 29, 40}, {25, 34, 39}, {24, 37, 37}} with same area 420 and same perimeter 98.

%e a(6) = 20026  because there exists 6 primitive Heron triangles: {{2108,8493,9425}, {2173,8398,9455}, {2261,8277,9488},{2418,8075,9533},{4123,6205,9698},{4588,5729,9709}} with same area 8410920 and same perimeter 20026.

%e a(8) = 721786  because there exists 8 primitive Heron triangles: {{188105,189428,344253}, {179133,198458,344195},{124338,256523,340925}, {116093,266029,339664}, {91448,299013,331325}, {88253,304980,328553}, {85981,310493,325312}, {85618,311627,324541}} with same area 13338605280 and same perimeter 721786.

%t sol=Association[];

%t For[n=6,n<=3000,n+=2,For[z=Ceiling[n/3],z<Floor[n/2],z++,For[x=1,x<Floor[n/3],x++,y=n-x-z;

%t If[x+y>z>=y>=x&&GCD[x,y,z]==1,p=(x+y+z)/2;A=Sqrt[p (p-x) (p-y) (p-z)];

%t If[IntegerQ[A],d=ToString@n<>"->"<>ToString@A;t={x,y,z};

%t If[KeyExistsQ[sol,d]==False,sol[d]={d}];

%t If[KeyExistsQ[sol,d],AppendTo[sol[d],t]]]]]]];

%t Do[Print[k,SelectFirst[sol,Length@#==k+1&]],{k,5}]

%o (Python)

%o from math import gcd

%o from collections import Counter

%o from itertools import count

%o from sympy.ntheory.primetest import is_square

%o def A385976(n):

%o     for k in count(6,2):

%o         c = Counter()

%o         for x in range(1,k//3+1):

%o             for y in range(x,k//2+1):

%o                 s, m = k>>1, x+y

%o                 if  (m<<1)>k>=m+y and gcd(x,y,k-m)==1 and is_square(s*(a:=(s-x)*(s-y)*(m-s))):

%o                     c[a]+=1

%o                     if c[a]>n:

%o                         break

%o             else:

%o                 continue

%o             break

%o         else:

%o             if any(c[d]==n for d in c):

%o                 return k # _Chai Wah Wu_, Jul 25 2025

%Y Cf. A385819.

%K nonn,hard,more

%O 1,1

%A _Zhining Yang_, Jul 13 2025

%E a(7)-a(8) from _Xianwen Wang_, Jul 13 2025

%E a(3) corrected by _Xianwen Wang_, Aug 19 2025