A106499 Ordered perimeter of primitive integer-sided triangle one of whose angles is twice another.
15, 28, 40, 45, 66, 77, 84, 91, 104, 120, 126, 144, 153, 170, 187, 190, 198, 209, 220, 228, 231, 260, 273, 276, 286, 299, 312, 322, 325, 345, 350, 378, 390, 400, 420, 432, 435, 442, 459, 464, 476, 493, 496, 510, 522, 527, 544, 551, 558, 561, 570, 589, 608
Offset: 1
Keywords
Links
- Jean-François Alcover, Table of n, a(n) for n = 1..869
Programs
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Mathematica
s[p_] := Solve[1 < r < s < 2r && p == r s + s^2 && GCD[r, s] == 1, {r, s}, Integers]; Reap[For[p = 1, p <= 10000, p++, sp = s[p]; If[sp =!= {}, Sow[p]; Print[p, " ", {r s, s^2 - r^2, r^2} /. sp[[1]]]]]][[2, 1]] (* Jean-François Alcover, Mar 06 2020 *)
A106420 Length of side opposite the greater of the two angles, one being the double of the other, of a primitive integer-sided triangle, sorted on perimeter (A106499).
6, 12, 15, 20, 30, 28, 35, 42, 40, 56, 45, 63, 72, 70, 66, 90, 77, 88, 99, 84, 110, 91, 104, 132, 117, 130, 143, 126, 156, 120, 154, 182, 165, 144, 195, 176, 210, 153, 170, 208, 187, 204, 240, 221, 198, 238, 255, 190, 234, 272, 209, 228, 247, 220, 266, 306, 285
Offset: 1
Keywords
Programs
-
Mathematica
sol[p_] := Solve[1 < r < s < 2 r && p == r s + s^2 && GCD[r, s] == 1, {r, s}, Integers]; Reap[For[p = 1, p <= 1000, p++, sp = sol[p]; If[sp =!= {}, Print[r s /. sp[[1]]]; Sow[r s /. sp[[1]]]]]][[2, 1]] (* Jean-François Alcover, Mar 07 2020 *)
A106506 Length of side common to the two angles, one being the double of the other, of a primitive integer-sided triangle, sorted on perimeter (A106499).
5, 7, 16, 9, 11, 33, 24, 13, 39, 15, 56, 32, 17, 51, 85, 19, 72, 57, 40, 95, 21, 120, 105, 23, 88, 69, 48, 115, 25, 161, 75, 27, 104, 175, 56, 135, 29, 208, 189, 87, 168, 145, 31, 120, 203, 93, 64, 261, 155, 33, 240, 217, 192, 279, 165, 35, 136, 231, 105, 320, 72, 37, 272
Offset: 1
Keywords
Programs
-
Mathematica
sol[p_] := Solve[1 < r < s < 2 r && p == r s + s^2 && GCD[r, s] == 1, {r, s}, Integers]; Reap[For[p = 1, p <= 1000, p++, sp = sol[p]; If[sp =!= {}, Print[s^2 - r^2 /. sp[[1]]]; Sow[s^2 - r^2 /. sp[[1]]]]]][[2, 1]] (* Jean-François Alcover, Mar 07 2020 *)
Extensions
Extended by Ray Chandler, May 09 2005
A147775 Numbers of the form (h+k)k, where h/k is a fraction in a Farey sequence, that occur as at least two times in a Farey sequence. 420 (13+15)*15=(1+20)*20.
420, 840, 858, 1260, 1326, 1540, 1716, 1938, 1980, 2145, 2310, 2340, 2520, 2622, 2652, 2730, 2850, 2860, 2964, 3080, 3450, 3465, 3740, 3876, 3960, 4002, 4080, 4290, 4350, 4420, 4560, 4620, 4650, 4692, 4845, 4940, 5005, 5100, 5244, 5394, 5460, 5544
Offset: 1
Keywords
Comments
This value (h+k)k gives one side of a triangle defined by sides (k^2, (h+k)k, (h+k)^2-k^k) which happens to have the property that the angle opposite the (h+k)k side is twice the measure of the angle opposite the k^2 side. These numbers would be the duplicates in the term-by-term multiplication: A106501 * A106500.
Examples
a(n) <- (h1,k1),(h2,k2) 420 <- (13,15),(1,20) 840 <- (19,21),(11,24) 858 <- (17,22),(7,26) 2160 <- (17,28),(1,35) 2520 <- (23,40),(11,45)
Links
- N. M. Makin, Table of n, a(n) for n = 1..5490.
Programs
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Mathematica
F[n_]:= Union[Flatten[Join[Table[p/q, {q, n}, {p, q - 1}]]]]; S[n_]:=(Denominator[n]+Numerator[n])Denominator[n]; FindDups[l_]:= Module[{sl,rs},sl=Sort[l];rs={};Map[If[Count[sl,# ]>1,rs=Append[rs,# ]]&,l]; Union[rs]]; FindDups[S[F[71]]]
Comments