A024365 -id:A024365 - cp's OEIS frontend

A057229 a(n) = ab = xy with (a-b) = (x+y) = A020882(n) (a>b, a>0, b>0, x>0, y>0), gcd(a, b) = gcd(x, y) = 1.

6, 30, 60, 84, 210, 210, 180, 630, 330, 504, 924, 1320, 546, 1386, 1560, 2340, 990, 2730, 840, 2574, 4620, 1224, 1716, 3570, 5610, 7140, 4290, 1710, 5016, 7956, 7980, 2730, 7854, 10374, 2310, 11970, 6630, 10920, 12540, 4080, 3036, 11856, 8970

Offset: 0

Naohiro Nomoto, Sep 19 2000

nonn

The quadratics in X, X^2 - S*X -+ P, where S=A020882(n), P=A057229(n) are each factorizable into two factors, all four being distinct: X^2 - S*X - P = (X - a)*(X + b). X^2 - S*X + P = (X - x)*(X - y). - Lekraj Beedassy, Apr 30 2004

Areas of primitive Pythagorean triangles sorted on hypotenuse A020882, then on perimeter A093507. - Lekraj Beedassy, Aug 18 2006

			E.g. a(1)=6=6*1=3*2, (6-1)=(3+2)=5=A020882(1), gcd(6,1)=gcd(3,2)=1

P. Yiu, Factorizable x^2 + px -+ q, Recreational Mathematics, pp. 58/360.

Cf. A020882, A008846, A024406, A024365.

A228874 a(n) = L(n) * L(n+1) * L(n+2) * L(n+3), the product of four consecutive Lucas numbers, A000032.

Original entry on oeis.org

24, 84, 924, 5544, 40194, 269874, 1864584, 12741324, 87431844, 599001144, 4106310474, 28143249834, 192901471224, 1322153872644, 9062210132844, 62113226746824, 425730613530834, 2918000448971874, 20000274149827944, 137083914357154044, 939587137457703924

Offset: 0

T. D. Noe, Sep 24 2013

Mohanty and Mohanty prove in Corollary 2.6 that these numbers are Pythagorean. The number a(n) is primitive Pythagorean if Lucas(n) and Lucas(n+1) have opposite parity. Every third number, starting at a(1) = 84, is not primitive Pythagorean.

Since a(n) = L(n+1)*L(n+2)*(L(n+2)^2-L(n+1)^2), these numbers are in A073120, - Robert Israel, Apr 06 2015

Robert Israel, Table of n, a(n) for n = 0..1193
Supriya Mohanty and S. P. Mohanty, Pythagorean Numbers, Fibonacci Quarterly 28 (1990), 31-42.
Index entries for linear recurrences with constant coefficients, signature (5,15,-15,-5,1).

Cf. A000032 (Lucas numbers), A228873 (similar sequence for Fibonacci numbers).

Cf. A009112 (Pythagorean numbers), A024365, A073120.

L:= n -> 2*combinat:-fibonacci(n+1)-combinat:-fibonacci(n):
seq(mul(L(n+i),i=0..3),n=0..30); # Robert Israel, Apr 06 2015

Table[LucasL[n] LucasL[n+1] LucasL[n+2] LucasL[n+3], {n, 0, 25}]
Times@@@Partition[LucasL[Range[0,30]],4,1] (* Harvey P. Dale, Jul 11 2017 *)

Vec(6*(x^4-4*x^3-24*x^2+6*x-4)/((x-1)*(x^2-7*x+1)*(x^2+3*x+1)) + O(x^100)) \\ Colin Barker, Oct 29 2013

G.f.: 6*(x^4-4*x^3-24*x^2+6*x-4) / ((x-1)*(x^2-7*x+1)*(x^2+3*x+1)). - Colin Barker, Oct 29 2013
From Robert Israel, Apr 06 2015: (Start)
a(n+5) = 5*a(n+4) + 15*a(n+3) - 15*a(n+2) - 5*a(n+1) + a(n).
a(n) = -A228873(n+3) + 4*A228873(n+2) + 24*A228873(n+1) - 6*A228873(n) + 4*A228873(n-1) for n >= 2. (End)
Sum_{n>=0} 1/a(n) = (10 - 3*sqrt(5))/60. - Diego Rattaggi, Aug 16 2021

A094807 Numbers n such that primitive solutions for 1/n^2 = 1/x^2 + 1/y^2 exist.

Original entry on oeis.org

12, 60, 120, 168, 360, 420, 660, 1008, 1092, 1260, 1680, 1848, 1980, 2448, 2640, 2772, 3120, 3420, 3432, 4620, 4680, 5148, 5460, 6072, 7140, 7800, 8160, 8580, 9240, 9828, 10032, 11220, 11628, 12180, 13260, 14280, 14880, 15708, 15912, 15960, 17940

Offset: 1

Lekraj Beedassy, Jun 11 2004

nonn

Numbers n that are the product of two legs of a primitive Pythagorean triangle, that is, n = 2xy(x^2-y^2) where x and y are two relatively prime positive integers of different parity and x is greater than y.

Numbers n which are the length of the altitude on the hypotenuse of a Pythagorean triangle and the smallest in its similarity class.

			12 is in the sequence because we have 1/12^2 = 1/15^2 + 1/20^2 and gcd(12,15,20)=1.

E. Bahier, Recherche Methodique et Proprietes des Triangles Rectangles en Nombres Entiers, Hermann, Paris, 1916. p. 68.

Equals 2*A024365(n).

Comments provided by Michael Somos, Oct 01 2004

A101439 Areas of primitive Pythagorean triangles which are palindromes.

Original entry on oeis.org

6, 63336, 474474, 666666, 4383834, 43177134, 460962269064, 60471399317406, 60490233209406, 643869171968346, 6708875775788076, 44703479297430744, 608857707707758806, 44523865177156832544, 683665820959028566386

Offset: 1

Zak Seidov, Jan 18 2005

Other parts of the n-th triangle are {m,n}, {a,b,c}:
a(1): {1,2}, {3,4,5};
a(2): {8,21}, {377,336,505};
a(3): {1,78}, {6083,156,6085};
a(4): {26,37}, {693,1924,2045};
a(5): {49,62}, {1443,6076,6245};
a(6): {11,158}, {24843,3476,25085};
a(7): {2376,2393}, {81073,11371536,11371825};
a(8): {4569,4858}, {2724403,44392404,44475925};
a(9): {2974,3773}, {5390853,22441804,23080205};
a(10): {5402,6829}, {17453637,73780516,75816845};
a(11): {121,38132}, {1454034783,9227944,1454064065};
a(12): {28407,29336}, {53643247,1666695504,1667558545};
a(13): {16593,35986}, {1019664547,1194231396,1570319845};
a(14): {3168,241339}, {58234476697,1529123904,58254549145};
a(15): {160034,213573}, {20002545173,68357882964,71224307485}.

			666666 is a member as it is a palindromic number and is the area of a primitive Pythagorean triangle with legs a=693 & b=1924 and hypotenuse c=2045.

Cf. A101450.

Cf. A002113, A024365.

lst = {}; Do[ If[ GCD[m, n] == 1, a = IntegerDigits[m*n^3 - n*m^3]; If[ Reverse[a] == a, lst = Sort[ AppendTo[ lst, a]]; Print[{n^2 - m^2, 2m*n, n^2 + m^2, m*n^3 - n*m^3}]]], {n, 55000}, {m, If[ EvenQ[n], 1, 2], n - 1, 2}]; lst (* Robert G. Wilson v, Jan 25 2005 *)

for(n=2,oo, is_A024365(a=A002113(n)) && print1(a", ")) \\ Could be made to a function returning, e.g., the n-th row := the n-digit terms. - M. F. Hasler, Jun 06 2024

Intersection of A002113 and A024365. - M. F. Hasler, Jun 06 2024

a(8) & a(10) - a(13) from Robert G. Wilson v, Jan 25 2005

a(14) and a(15) from Ray Chandler, Feb 10 2013

A227201 Areas of Pythagorean triangles with areas that use all ten decimal digits exactly once.

Original entry on oeis.org

1072493856, 1075948326, 1235976840, 1239084756, 1253684790, 1253894670, 1263984750, 1265738940, 1270953864, 1279635840, 1287349560, 1287463950, 1295738640, 1297836540, 1298647350, 1328405976, 1329754860, 1346589720, 1357962840, 1376925480, 1382974560

Offset: 1

Charles Kluepfel, Sep 18 2013

There are 515 members to the set.

T. D. Noe, Table of n, a(n) for n = 1..515

Cf. A024365.

mx = 10^10; mn = mx/10; nn = Ceiling[mx^(1/3)]; If[OddQ[nn], nn + 1]; t = Union[Flatten[Table[If[GCD[u, v] == 1 && Mod[u, 2] + Mod[v, 2] == 1, u v (u^2 - v^2), 0], {u, nn}, {v, u - 1}]]]; t2 = Select[Rest[t], # < mx &]; t3 = {}; Do[num0 = Ceiling[Sqrt[mn/n]]; num1 = Floor[Sqrt[mx/n]]; Do[num = i^2*n; If[Length[Union[IntegerDigits[num]]] == 10, AppendTo[t3, num]], {i, num0, num1}], {n, t2}]; t3 = Union[t3] (* T. D. Noe, Sep 20 2013 *)

A258151 Areas of primitive Pythagorean triangles divided by 6, in increasing order without multiple entries.

Original entry on oeis.org

1, 5, 10, 14, 30, 35, 55, 84, 91, 105, 140, 154, 165, 204, 220, 231, 260, 285, 286, 385, 390, 429, 455, 506, 595, 650, 680, 715, 770, 819, 836, 935, 969, 1015, 1105, 1190, 1240, 1309, 1326, 1330, 1495, 1496, 1615, 1729, 1771, 1785, 1820, 1925

Offset: 1

Wolfdieter Lang, Jun 14 2015

See A020885 for this sequence with multiplicities. See A024365 for the areas with multiplicities.

This sequence gives also Fibonacci's congruous numbers divided by 24 without multiple entries. See A258150.

			See A020885.

Cf. A020885, A024365, A258150.

cp's OEIS Frontend

A057229 a(n) = a*b = x*y with (a-b) = (x+y) = A020882(n) (a>b, a>0, b>0, x>0, y>0), gcd(a, b) = gcd(x, y) = 1.

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A228874 a(n) = L(n) * L(n+1) * L(n+2) * L(n+3), the product of four consecutive Lucas numbers, A000032.

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A094807 Numbers n such that primitive solutions for 1/n^2 = 1/x^2 + 1/y^2 exist.

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A101439 Areas of primitive Pythagorean triangles which are palindromes.

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Mathematica

PARI

Formula

Extensions

A227201 Areas of Pythagorean triangles with areas that use all ten decimal digits exactly once.

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Programs

Mathematica

A258151 Areas of primitive Pythagorean triangles divided by 6, in increasing order without multiple entries.

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Comments

Examples

Crossrefs

A057229 a(n) = ab = xy with (a-b) = (x+y) = A020882(n) (a>b, a>0, b>0, x>0, y>0), gcd(a, b) = gcd(x, y) = 1.