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Perimeters of Heronian triangles whose smallest side length is coprime to their middle side and whose middle side length divides their perimeter.
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%I #16 Feb 16 2025 08:34:00

%S 12,42,78,84,114,156,186,222,228,258,294,366,372,402,438,444,474,516,

%T 546,582,588,618,654,732,762,798,804,834,876,906,942,948,978,1014,

%U 1086,1092,1158,1164,1194,1236,1266,1302,1308,1338,1374,1446,1482,1524,1554,1596

%N Perimeters of Heronian triangles whose smallest side length is coprime to their middle side and whose middle side length divides their perimeter.

%C Among the 42352 terms below 2*10^6 we observe that if multiples triangles correspond to a term their middle sides coincide. In this range there are perimeters common to 1, 2, 4, 8, and 16 triangles. The first such terms are a(1) = 12, a(4) = 84, a(36) = 1092, a(546) = 20748, a(14273) = 643188, and the corresponding middle sides are 4, 28, 364, 6916, and 214396. - _Giovanni Resta_, May 30 2020

%H Giovanni Resta, <a href="/A335150/b335150.txt">Table of n, a(n) for n = 1..10000</a>

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

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Heronian_triangle">Heronian triangle</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Integer_triangle">Integer Triangle</a>

%Y Cf. A335149.

%K nonn,changed

%O 1,1

%A _Wesley Ivan Hurt_, May 25 2020

%E Terms a(27) and beyond from _Giovanni Resta_, May 30 2020