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 A229159 Smallest integer areas of integer-sided triangles where at least one side is of length prime(n). 2
 0, 6, 6, 42, 66, 24, 36, 114, 966, 60, 930, 114, 126, 1290, 4230, 90, 1770, 330, 2814, 14910, 216, 4740, 1494, 420, 420, 510, 6180, 4494, 840, 570, 8382, 11790, 630, 9174, 210, 4530, 840, 2934, 45090, 3276, 22554, 1260, 24066, 336, 1386, 16716, 26586, 52182 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Conjecture: for all prime p > 2 there exists an integer-sided triangle with integer area where at least one side is of length p. There exist triangles of integer area and integer side lengths having two sides whose lengths are distinct prime numbers; for example, (3,4,5), (11,13,20), (19, 20,37), (43,61,68), (59,68,109), (11,60,61), (79,241, 312), (41,50,89), (26,73,97), ... corresponding to the areas 6, 66, 114, 1290, 1770, 330, 4740, 420, 420, ... Observation: there exist some integer-area, integer-sided triangles with two prime sides such that the perimeter equals 4 times the smaller prime. For example: (3, 4, 5) => 12 = 4*3; (11, 13, 20) => 44 = 4*11; (19, 20, 37) => 76 = 4*19; (43, 61, 68) => 172 = 4*43; (59, 68, 109) => 236 = 4*59; (131, 181, 212) => 524 = 4*131; (139, 157, 260) => 556 = 4*139; (179, 260, 277) => 716 = 4*179. The first 25 values (prime(n), smallest area, a, b, c) are: +---------+-------+-----+-----+-----+ | prime(n)| Area | a | b | c | +---------+-------+-----+-----+-----+ | 2 | 0 | 0 | 0 | 0 | | 3 | 6 | 3 | 4 | 5 | | 5 | 6 | 3 | 4 | 5 | | 7 | 42 | 7 | 15 | 20 | | 11 | 66 | 11 | 13 | 20 | | 13 | 24 | 4 | 13 | 15 | | 17 | 36 | 9 | 10 | 17 | | 19 | 114 | 19 | 20 | 37 | | 23 | 966 | 23 | 140 | 159 | | 29 | 60 | 6 | 25 | 29 | | 31 | 930 | 31 | 68 | 87 | | 37 | 114 | 19 | 20 | 37 | | 41 | 126 | 15 | 28 | 41 | | 43 | 1290 | 43 | 61 | 68 | | 47 | 4230 | 47 | 425 | 468 | | 53 | 90 | 4 | 51 | 53 | | 59 | 1770 | 59 | 68 | 109 | | 61 | 330 | 11 | 60 | 61 | | 67 | 2814 | 67 | 85 | 116 | | 71 | 14910 | 71 | 447 | 476 | | 73 | 216 | 9 | 73 | 80 | | 79 | 4740 | 79 | 241 | 312 | | 83 | 1494 | 83 | 85 | 164 | | 89 | 420 | 41 | 50 | 89 | | 97 | 420 | 26 | 73 | 97 | LINKS Table of n, a(n) for n=1..48. MAPLE with(numtheory):nn:=500: for m from 2 to 40 do: q:=ithprime(m):ii:=0:for a from 1 to nn while(ii=0) do: for b from a to nn while(ii=0) do: for c from b to nn while(ii=0) do: p:=(a+b+c)/2 : x:=p*(p-a)*(p-b)*(p-c): if x>0 then x0:= sqrt(x):else fi:if (x0=floor(x0) and a=q) or (x0=floor(x0) and b=q) or (x0=floor(x0) and c=q)then ii:=1: printf ( "%d %d %d %d %d \n", q, x0, a, b, c): :else fi:od:od:od:od: CROSSREFS Cf. A226453. Sequence in context: A279535 A125510 A117859 * A102901 A014435 A175550 Adjacent sequences: A229156 A229157 A229158 * A229160 A229161 A229162 KEYWORD nonn AUTHOR Michel Lagneau, Sep 17 2013 STATUS approved

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Last modified May 20 03:57 EDT 2024. Contains 372703 sequences. (Running on oeis4.)