A385976 Least integer k such that there are exactly n primitive Heron triangles having the same area and perimeter k.
12, 70, 98, 2002, 2842, 20026, 91698, 721786
Offset: 1
Examples
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.
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.
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.
Links
- BBS Math Blog, Primitive Heron triangles with equal perimeter and area (in Chinese).
Crossrefs
Cf. A385819.
Programs
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Mathematica
sol=Association[]; For[n=6,n<=3000,n+=2,For[z=Ceiling[n/3],z
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Python
from math import gcd from collections import Counter from itertools import count from sympy.ntheory.primetest import is_square def A385976(n): for k in count(6,2): c = Counter() for x in range(1,k//3+1): for y in range(x,k//2+1): s, m = k>>1, x+y if (m<<1)>k>=m+y and gcd(x,y,k-m)==1 and is_square(s*(a:=(s-x)*(s-y)*(m-s))): c[a]+=1 if c[a]>n: break else: continue break else: if any(c[d]==n for d in c): return k # Chai Wah Wu, Jul 25 2025
Extensions
a(7)-a(8) from Xianwen Wang, Jul 13 2025
a(3) corrected by Xianwen Wang, Aug 19 2025