A336900 - OEIS

%I #17 Aug 08 2020 04:32:13

%S 4,5,9,10,17,15,20,20,26,25,34,30,40,35,36,40,41,45,52,50,52,55,58,60,

%T 68,65,72,65,73,72,85,80,85,82,98,89,101,95,100,100,104,104,101,110,

%U 106,115,113,117,130,125,123,128,130,135,136,130,145,145,164,144

%N a(n) is the first element of the lexicographically earliest Heron square root triple with area n > 0, or 0, if no such triple exists.

%C A Heron square root triple is a triple of positive integers [a, b, c] with a >= b >= c such that Heron's formula applied to their square roots leads an integer value. This means: H(a, b, c) = sqrt(S*(S - A)*(S - B)*(S - C)) where A = sqrt(a), B = sqrt(b), C = sqrt(c) and S = (A + B + C)/2. We call H(a, b, c) the area of the triple [a, b, c].

%C _Samuel Bodansky_ introduced Heron square root triple in A334818. The term 'Heron square root triple' is not standard.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Heron%27s_formula">Heron's formula</a>.

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

%e a(1) = 4 because H[4, 2, 2] = sqrt((sqrt(2) + 1)*(sqrt(2) - 1)) = 1.

%e a(40) = 100 because H[100, 89, 89] = sqrt(25*(5 + sqrt(89))*(-5 + sqrt(89))) = sqrt(1600) = 40.

%e In both cases there are also no lexicographically earlier Heron square root triple that take this value.

%e .

%e The first few triples are:

%e [ 4,  2,  2] ->  1;   [40, 29, 25] -> 13;   [68, 58, 50] -> 25;

%e [ 5,  5,  4] ->  2;   [35, 35, 28] -> 14;   [65, 65, 52] -> 26;

%e [ 9,  8,  5] ->  3;   [36, 34, 34] -> 15;   [72, 65, 53] -> 27;

%e [10, 10,  8] ->  4;   [40, 40, 32] -> 16;   [65, 65, 64] -> 28;

%e [17, 13,  8] ->  5;   [41, 40, 37] -> 17;   [73, 68, 61] -> 29;

%e [15, 15, 12] ->  6;   [45, 45, 36] -> 18;   [72, 68, 68] -> 30;

%e [20, 17, 13] ->  7;   [52, 50, 34] -> 19;   [85, 68, 65] -> 31;

%e [20, 20, 16] ->  8;   [50, 50, 40] -> 20;   [80, 80, 64] -> 32;

%e [26, 20, 18] ->  9;   [52, 49, 45] -> 21;   [85, 73, 72] -> 33;

%e [25, 25, 20] -> 10;   [55, 55, 44] -> 22;   [82, 80, 74] -> 34;

%e [34, 26, 20] -> 11;   [58, 52, 50] -> 23;   [98, 82, 68] -> 35;

%e [30, 30, 24] -> 12;   [60, 60, 48] -> 24;   [89, 81, 80] -> 36;

%Y Cf. A334818.

%K nonn

%O 1,1

%A _Peter Luschny_, Aug 07 2020