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 A322105 Numbers that set a record for occurrences as longest side of a triangle with integer sides and positive integer area. 3
 1, 5, 13, 15, 25, 30, 52, 65, 75, 100, 120, 145, 195, 300, 325, 390, 520, 585, 600, 650, 780, 975, 1105, 1300, 1560, 1700, 1950, 2550, 2600, 3315, 3900, 4420, 5100, 5525, 6630, 7800, 8840, 10200, 11050, 13260, 16575, 22100, 26520, 33150, 44200, 53040, 66300, 96135 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Congruent triangles are identified, that is to say mirror images are not distinguished. The corresponding numbers of occurrences are 0, 1, 2, 3, 4, 7, 10, ... A054875(k) gives the number of occurrences for any integer k. LINKS Ray Chandler, Table of n, a(n) for n = 1..79 (terms < 6*10^6; first 67 terms from Giovanni Resta) Ray Chandler, First 79 terms with corresponding occurrences (first 67 terms from Giovanni Resta) EXAMPLE 13 is in the sequence since it occurs in a record number of 2 triangles of side lengths {5, 12, 13} and {10, 13, 13}. The side lengths, a(n), and their corresponding record numbers of occurrences, A054875(a(n)), are: n a(n) prime factorization of a(n) occurrences 1 1 - 0 2 5 5 1 3 13 13 2 4 15 3 * 5 3 5 25 5^2 4 6 30 2 * 3 * 5 7 7 52 2^2 * 13 10 8 65 5 * 13 11 9 75 3 * 5^2 13 10 100 2^2 * 5^2 15 11 120 2^3 * 3 * 5 22 12 145 5 * 29 23 13 195 3 * 5 * 13 35 14 300 2^2 * 3 * 5^2 41 15 325 5^2 * 13 51 16 390 2 * 3 * 5 * 13 57 17 520 2^3 * 5 * 13 63 18 585 3^2 * 5 * 13 64 19 600 2^3 * 3 * 5^2 72 20 650 2 * 5^2 * 13 82 21 780 2^2 * 3 * 5 * 13 94 22 975 3 * 5^2 * 13 135 23 1105 5 * 13 * 17 143 24 1300 2^2 * 5^2 * 13 158 25 1560 2^3 * 3 * 5 * 13 171 26 1700 2^2 * 5^2 * 17 182 27 1950 2 * 3 * 5^2 * 13 210 28 2550 2 * 3 * 5^2 * 17 216 29 2600 2^3 * 5^2 * 13 251 30 3315 3 * 5 * 13 * 17 333 31 3900 2^2 * 3 * 5^2 * 13 367 32 4420 2^2 * 5 * 13 * 17 373 33 5100 2^2 * 3 * 5^2 * 17 406 34 5525 5^2 * 13 * 17 496 35 6630 2 * 3 * 5 * 13 * 17 525 36 7800 2^3 * 3 * 5^2 * 13 605 37 8840 2^3 * 5 * 13 * 17 610 38 10200 2^3 * 3 * 5^2 * 17 660 39 11050 2 * 5^2 * 13 * 17 735 40 13260 2^2 * 3 * 5 * 13 * 17 897 41 16575 3 * 5^2 * 13 * 17 1132 42 22100 2^2 * 5^2 * 13 * 17 1276 MATHEMATICA okQ[x_, y_, z_] := If[x + y <= z, False, Module[{s = (x + y + z)/2}, IntegerQ[ Sqrt[s(s-x)(s-y)(s-z)]]] ]; a[n_] := Module[{num = 0}, Do[Do[If[okQ[x, y, n], num++], {x, 1, y}], {y, 1, n}]; num]; amax=-1; s={}; Do[a1=a[n]; If[a1 > amax, AppendTo[s, n]; amax=a1], {n, 1, 100}]; s CROSSREFS Cf. A054875, A096467, A120130, A306626. Sequence in context: A327922 A309812 A324909 * A230503 A335695 A067696 Adjacent sequences: A322102 A322103 A322104 * A322106 A322107 A322108 KEYWORD nonn AUTHOR Amiram Eldar and Peter Munn, Nov 26 2018 EXTENSIONS a(43)-a(48) from Giovanni Resta, Nov 03 2019 STATUS approved

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Last modified December 6 08:34 EST 2023. Contains 367600 sequences. (Running on oeis4.)