A073120 -id:A073120 - cp's OEIS frontend

A057102 a(n) = 4 * A073120(n).

24, 96, 120, 240, 336, 384, 480, 720, 840, 960, 1320, 1344, 1536, 1920, 1944, 2016, 2184, 2520, 2880, 3360, 3696, 3840, 3960, 4896, 5280, 5376, 5544, 6144, 6240, 6840, 6864, 7680, 7776, 8064, 8736, 9240, 9360, 9720, 10080, 10296, 10920, 11520, 12144

Offset: 1

Henry Bottomley, Aug 02 2000

This sequence was originally described as the list of "congrua". But that name more properly refers to A256418.

Numbers of the form 4*(x^3*y-x*y^3) (where x,y are integers and x>=y). Squares of these numbers are of the form N^4-K^2 (where N belongs to A135786 and K to A135789 or A135790). Proof uses identity: (4(x^3y-xy^3))^2=(x^2+y^2)^4-(x^4 - 6x^2 y^2 + y^4)^2. - Artur Jasinski, Nov 29 2007, Nov 14 2008

Cf. A073120, A256418.

N:= 10^5: # to get all terms <= N
select(`<=`,{seq(seq(4*(x^3*y-x*y^3),y=1..x-1),x=1..floor(sqrt(N/4+1)))},N);
# If using Maple 11 or earlier, uncomment the following line
# sort(convert(%, list)); # Robert Israel, Apr 06 2015

a = {}; Do[Do[w = 4x^3y - 4x y^3; If[w > 0 && w < 10000, AppendTo[a, w]], {x, y, 1000}], {y, 1, 1000}]; Union[a] (* Artur Jasinski, Nov 29 2007 *)

Edited by N. J. A. Sloane, Apr 06 2015 at the suggestion of Robert Israel, Apr 03 2015

A003273 Congruent numbers: positive integers k for which there exists a right triangle having area k and rational sides.

Original entry on oeis.org

5, 6, 7, 13, 14, 15, 20, 21, 22, 23, 24, 28, 29, 30, 31, 34, 37, 38, 39, 41, 45, 46, 47, 52, 53, 54, 55, 56, 60, 61, 62, 63, 65, 69, 70, 71, 77, 78, 79, 80, 84, 85, 86, 87, 88, 92, 93, 94, 95, 96, 101, 102, 103, 109, 110, 111, 112, 116, 117, 118, 119, 120, 124, 125, 126

Offset: 1

N. J. A. Sloane

Positive integers k such that x^2 + k*y^2 = z^2 and x^2 - k*y^2 = t^2 have simultaneous integer solutions. In other words, k is the difference of an arithmetic progression of three rational squares: (t/y)^2, (x/y)^2, (z/y)^2. Values of k corresponding to y=1 (i.e., an arithmetic progression of three integer squares) form A256418.
Tunnell shows that if a number is squarefree and congruent, then the ratio of the number of solutions of a pair of equations is 2. If the Birch and Swinnerton-Dyer conjecture is assumed, then determining whether a squarefree number k is congruent requires counting the solutions to a pair of equations. For odd k, see A072068 and A072069; for even k see A072070 and A072071.
If a number k is congruent, there are an infinite number of right triangles having rational sides and area k. All congruent numbers can be obtained by multiplying a primitive congruent number A006991 by a positive square number A000290.
Conjectured asymptotics (based on random matrix theory) on p. 453 of Cohen's book. - Steven Finch, Apr 23 2009

			24 is congruent because 24 is the area of the right triangle with sides 6,8,10.
5 is congruent because 5 is the area of the right triangle with sides 3/2, 20/3, 41/6 (although not of any right triangle with integer sides -- see A073120). - _Jonathan Sondow_, Oct 04 2013

Alter, Ronald; Curtz, Thaddeus B.; Kubota, K. K. Remarks and results on congruent numbers. Proceedings of the Third Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1972), pp. 27-35. Florida Atlantic Univ., Boca Raton, Fla., 1972. MR0349554 (50 #2047)
H. Cohen, Number Theory. I, Tools and Diophantine Equations, Springer-Verlag, 2007, p. 454. [From Steven Finch, Apr 23 2009]
R. Cuculière, "Mille ans de chasse aux nombres congruents", in Pour la Science (French edition of 'Scientific American'), No. 7, 1987, pp. 14-18.
L. E. Dickson, History of the Theory of Numbers, Vol. 2, pp. 459-472, AMS Chelsea Pub. Providence RI 1999.
R. K. Guy, Unsolved Problems in Number Theory, D27.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Congruent numbers up to 10000; table of n, a(n) for n = 1..5742
R. Alter, Letter to N. J. A. Sloane, Sep 1975
R. Alter, The congruent number problem, Amer. Math. Monthly, 87 (1980), 43-45.
R. Alter and T. B. Curtz, A note on congruent numbers, Math. Comp., 28 (1974), 303-305 and 30 (1976), 198.
A. Alvarado and E. H. Goins, Arithmetic progressions on conic sections, arXiv:1210.6612 [math.NT], 2012. [From _Jonathan Sondow_, Oct 25 2012]
Estelle Basor and Bill Hart, A trillion triangles, American Institute of Mathematics,
E. Brown, Three Fermat Trails to Elliptic Curves, 5. Congruent Numbers and Elliptic Curves (pp 8-11/17)
Graeme Brown, The Congruent Number Problem, 2014.
Jasbir S. Chahal, Some remarks on rational right triangles, Expos. Math. (2024).
B. Cipra, Tallying the class of congruent numbers, ScienceNOW, Sep 23 2009.
Clay Mathematics Institute, The Birch and Swinnerton-Dyer Conjecture
Raiza Corpuz, Constructing congruent number elliptic curves using 2-descent, arXiv:2006.08113 [math.NT], 2020.
R. Cuculière, Mille ans de chasse aux nombres congruents, Séminaire de Philosophie et Mathématiques, 2, 1988, p. 1-17.
Department of Pure Maths., Univ. Sheffield, Pythagorean triples and the congruent number problem [Cached copy]
A. Dujella, A. S.Janfeda, and S. Salami, A Search for High Rank Congruent Number Elliptic Curves, JIS 12 (2009) 09.5.8.
E. V. Eikenberg, The Congruent Number Problem
David Goldberg, Triangle Sides for Congruent Numbers less than 10000, arXiv:2106.07373 [math.NT], 2021.
Lorenz Halbeisen and Norbert Hungerbühler, Congruent number elliptic curves with rank at least two, arXiv:1809.02037 [math.NT], 2018.
Lorenz Halbeisen and Norbert Hungerbühler, Congruent Number Elliptic Curves Related to Integral Solutions of m^2 = n^2 + nl + n^2, J. Int. Seq., Vol. 22 (2019), Article 19.3.1.
Alvaro Lozano-Robledo, My #MegaFavNumber: 224,403,517,704,336,969,924,557,513,090,674,863,160,948,472,041, video (2020) [discusses congruent numbers and a(157)]
W. F. Hammond, A Reading of Karl Rubin's SUMO Slides on Rational Right Triangles and Elliptic Curves
Bill Hart, A Trillion Triangles, American Institute of Mathematics.
T. Komatsu, Congruent numbers and continued fractions, Fib. Quart., 50 (2012), 222-226. - From _N. J. A. Sloane_, Mar 04 2013
S. Komoto, T. Watanabe and H. Wada, 42553 is a congruent number.
G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 (1986), 337-340.
G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 (1986), 337-340. [Annotated, corrected, scanned copy]
Allan J. MacLeod, The congruent number descent of Komotu, Watanabe and Wada, arXiv:2005.02615 [math.NT], 2020.
MathDL, Five Mathematicians Capture Record Number of Congruent Numbers
Fidel Ronquillo Nemenzo, All congruent numbers less than 40000, Proc. Japan Acad. Ser. A Math. Sci., Volume 74, Number 1 (1998), 29-31.
Karl Rubin, Right triangles and elliptic curves
W. A. Stein, Introduction to the Congruent Number Problem
W. A. Stein, The Congruent Number Problem
Ye Tian, Congruent Numbers and Heegner Points, arXiv:1210.8231 [math.NT], 2012.
J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.
D. J. Wright, The Congruent Number Problem

Cf. A006991, A072068, A072069, A072070, A072071, A073120, A165564, A182429, A256418, A259680-A259687.

(* The following Mathematica code assumes the truth of the Birch and Swinnerton-Dyer conjecture and uses the list of primitive congruent numbers produced by the Mathematica code in A006991: *)
For[cLst={}; i=1, i<=Length[lst], i++, n=lst[[i]]; j=1; While[n j^2<=maxN, cLst=Union[cLst, {n j^2}]; j++ ]]; cLst

Guy gives a table up to 1000.
Edited by T. D. Noe, Jun 14 2002
Comments revised by Max Alekseyev, Nov 15 2008
Comment corrected by Jonathan Sondow, Oct 10 2013

A009112 Areas of Pythagorean triangles: numbers which can be the area of a right triangle with integer sides.

Original entry on oeis.org

6, 24, 30, 54, 60, 84, 96, 120, 150, 180, 210, 216, 240, 270, 294, 330, 336, 384, 480, 486, 504, 540, 546, 600, 630, 720, 726, 750, 756, 840, 864, 924, 960, 990, 1014, 1080, 1176, 1224, 1320, 1344, 1350, 1386, 1470, 1500, 1536, 1560, 1620, 1710, 1716, 1734, 1890

Offset: 1

David W. Wilson

Number of terms < 10^k for increasing values of k: 1, 7, 34, 150, 636, 2536, 9757, 35987, 125350, 407538, ..., .
All terms are divisible by 6.
Also positive integers m with four (or more) different divisors (p, q, r, s) such that m = p*q = r*s and s = p+q+r. - Jose Aranda, Jun 28 2023

			30 belongs to the sequence as the area of the triangle (5,12,13) is 30.
6 is in the sequence because it is the area of the 3-4-5 triangle.

Robert Israel, Table of n, a(n) for n = 1..10000
Ron Knott, Pythagorean Triangles
B. Miller, Nasty Numbers, The Mathematics Teacher 73 (1980), page 649.
Supriya Mohanty and S. P. Mohanty, Pythagorean Numbers, Fibonacci Quarterly 28 (1990), 31-42.

Union of A009111, A009127, A024365, A177021.
A073120 is a subsequence.
See A256418 for the numbers 4*a(n).

N:= 10^4: # to get all entries <= N
A:= {}:
for t from 1 to floor(sqrt(2*N)) do
   F:= select(f -> f[2]::odd,ifactors(2*t)[2]);
   d:= mul(f[1],f=F);
   for e from ceil(sqrt(t/d)) do
     s:= d*e^2;
     r:= sqrt(2*t*s);
     a:= (r+s)*(r+t)/2;
     if a > N then break fi;
     A:= A union {a};
   od
od:
A;
# if using Maple 11 or earlier, uncomment the next line
# sort(convert(A,list)); # Robert Israel, Apr 06 2015

lst = {}; Do[ If[ IntegerQ[c = Sqrt[a^2 + b^2]], AppendTo[lst, a*b/2]; lst = Union@ lst], {a, 4, 180}, {b, a - 1, Floor[ Sqrt[a]], -1}]; Take[lst, 51] (* Vladimir Joseph Stephan Orlovsky, Nov 23 2010 *)
g@A_ := With[{div = Divisors@(2*A)}, AnyTrue[Sqrt@(Plus@@({#, 2*A/#}^2))& /@Take[div, Floor[(Length@div)/2]],IntegerQ]];
Select[Range@5000, g@# &] (* Hans Rudolf Widmer, Sep 25 2023 *)

is_A009112(n)={ my(N=1+#n=divisors(2*n)); for( i=1, N\2, issquare(n[i]^2+n[N-i]^2) & return(1)) } \\ M. F. Hasler, Dec 09 2010

is_A009112 = lambda n: any(is_square(a**2+(2*n/a)**2) for a in divisors(2*n)) # D. S. McNeil, Dec 09 2010

A024365 Areas of right triangles with coprime integer sides.

Original entry on oeis.org

6, 30, 60, 84, 180, 210, 330, 504, 546, 630, 840, 924, 990, 1224, 1320, 1386, 1560, 1710, 1716, 2310, 2340, 2574, 2730, 3036, 3570, 3900, 4080, 4290, 4620, 4914, 5016, 5610, 5814, 6090, 6630, 7140, 7440, 7854, 7956, 7980, 8970, 8976, 9690, 10374

Offset: 1

David W. Wilson

Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A <= B); sequence gives areas A*B/2.
By Theorem 2 of Mohanty and Mohanty, all these numbers are primitive Pythagorean. - T. D. Noe, Sep 24 2013
This sequence also gives Fibonacci's congruous numbers (without multiplicity, in increasing order) divided by 4. See A258150. - Wolfdieter Lang, Jun 14 2015
The same as A024406 with duplicates removed. All terms are multiples of 6, cf. A258151. - M. F. Hasler, Jan 20 2019

			6 is in the sequence because it is the area of the 3-4-5 triangle.
a(7) = 210 corresponds to the two primitive Pythagorean triangles (21, 20, 29) and (35, 12, 37). See A024406. - _Wolfdieter Lang_, Jun 14 2015

T. D. Noe, Table of n, a(n) for n = 1..10000 (corrected by _Giovanni Resta_, Jan 21 2019)
Supriya Mohanty and S. P. Mohanty, Pythagorean Numbers, Fibonacci Quarterly 28 (1990), 31-42.

Cf. A009111, A009112, A024406 (with multiplicity), A258150, A024407, A258151 (terms divided by 6).

Subsequence of A073120 and A147778.

nn = 22; (* nn must be even *) 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}]]]; Select[Rest[t], # < nn (nn^2 - 1) &] (* T. D. Noe, Sep 19 2013 *)

select( {is_A024365(n)=my(N=1+#n=divisors(2*n)); for(i=1, N\2, gcd(n[i], n[N-i])==1 && issquare(n[i]^2+n[N-i]^2) && return(n[i]))}, [1..10^4]) \\ is_A024365 returns the smaller leg if n is a term, else 0. - M. F. Hasler, Jun 06 2024

Positive integers of the form u*v*(u^2 - v^2) where 2uv and u^2 - v^2 are coprime or, alternatively, where u, v are coprime and one of them is even.

a(n) = 6*A258151(n). - M. F. Hasler, Jan 20 2019

Additional comments James R. Buddenhagen, Aug 10 2008 and from Max Alekseyev, Nov 12 2008

Edited by N. J. A. Sloane, Nov 20 2008 at the suggestion of R. J. Mathar

A256418 Congrua (possible solutions to the congruum problem): numbers k such that there are integers x, y and z with k = x^2-y^2 = z^2-x^2.

Original entry on oeis.org

24, 96, 120, 216, 240, 336, 384, 480, 600, 720, 840, 864, 960, 1080, 1176, 1320, 1344, 1536, 1920, 1944, 2016, 2160, 2184, 2400, 2520, 2880, 2904, 3000, 3024, 3360, 3456, 3696, 3840, 3960, 4056, 4320, 4704, 4896, 5280, 5376, 5400, 5544

Offset: 1

N. J. A. Sloane, Apr 06 2015, following a suggestion from Robert Israel, Apr 03 2015

nonn

k is a "congruum" iff k/4 is the area of a Pythagorean triangle, so these are the numbers 4*A009112.
Each congruum is a multiple of 24; it cannot be a square.
This entry incorporates many comments that were originally in A057102. A057103 and A055096 need to be checked.

			a(11)=840 since 840=29^2-1^2=41^2-29^2 (indeed also 840=37^2-23^2=47^2-37^2).

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Congruum (but beware errors)
Wikipedia, Congruum (but beware errors).

Cf. A004431 for possible values of x in definition. Cf. A057103, A055096 for triangles of all congrua and values of x.

Cf. A009112, A073120, A135789, A135786.

r[n_] := Reduce[0 < y < x && 0 < x < z && n == x^2 - y^2 == z^2 - x^2, {x, y, z}, Integers];
Reap[For[n = 24, n < 10^4, n += 24, rn = r[n]; If[rn =!= False, Print[n, " ", rn]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Feb 25 2019 *)

A147778 Positive integers of the form uv(u^2 - v^2) where u, v are coprime integers.

Original entry on oeis.org

6, 24, 30, 60, 84, 120, 180, 210, 240, 330, 336, 504, 546, 630, 720, 840, 924, 990, 1224, 1320, 1386, 1560, 1710, 1716, 2016, 2184, 2310, 2340, 2520, 2574, 2730, 3036, 3360, 3570, 3696, 3900, 3960, 4080, 4290, 4620, 4896, 4914, 5016, 5280, 5544, 5610, 5814

Offset: 1

Max Alekseyev, Nov 12 2008

nonn

Terms with even u or v form A024365. Squarefree terms form A147779.

Robert Israel, Table of n, a(n) for n = 1..10000

Subsequence of: A003273, A009112, A073120.

N:= 10^5:
A:= {}:
for v from 1 to floor((N/2)^(1/3)) do
   for u from v+1 do
      if igcd(u,v) = 1 then
        t:= u*v*(u^2-v^2);
        if t > N then break fi;
        A:= A union {t};
      fi
    od
od:
A;
# if using Maple 11 or earlier, uncomment the next line
# sort(convert(A,list)); # Robert Israel, Apr 06 2015

A228873 a(n) = F(n) * F(n+1) * F(n+2) * F(n+3), the product of four consecutive Fibonacci numbers, A000045.

Original entry on oeis.org

6, 30, 240, 1560, 10920, 74256, 510510, 3495030, 23965920, 164237040, 1125770256, 7715953440, 52886430870, 362487682830, 2484530961360, 17029219589256, 116720030923320, 800010932051760, 5483356663145790, 37583485265670630, 257601041359736256

Offset: 1

T. D. Noe, Sep 24 2013

nonn

Mohanty and Mohanty prove in Corollary 2.5 that these numbers are Pythagorean. The number a(n) is primitive Pythagorean if F(n) and F(n+1) have opposite parity. Every third number, starting at a(1) = 6, is not primitive Pythagorean.

Since a(n) = F(n+1)*F(n+2)*(F(n+2)^2 - F(n+1)^2), a(n) is in A073120. - Robert Israel, Apr 06 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Supriya Mohanty and S. P. Mohanty, Pythagorean Numbers, The Fibonacci Quarterly, Vol. 28, No. 1 (1990), pp. 31-42.
H.-J. Seiffert, Problem B-1020, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 44, No. 4 (2006), p. 278; Two Fibonacci Identities, Solution to Problem B-1020 by Harris Kwong, ibid., Vol. 45, No. 2 (2007), pp. 182-184.
Index entries for linear recurrences with constant coefficients, signature (5,15,-15,-5,1).

Cf. A000045 (Fibonacci numbers), A228874 (similar sequence for Lucas numbers).

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

[Fibonacci(n)*Fibonacci(n+1)*Fibonacci(n+2)*Fibonacci(n+3): n in [1..30]]; // Vincenzo Librandi, Oct 04 2013

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

Table[Fibonacci[n] Fibonacci[n+1] Fibonacci[n+2] Fibonacci[n+3], {n, 25}]
CoefficientList[Series[-6/((x - 1) (x^2 + 3 x + 1) (x^2 - 7 x + 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 04 2013 *)
Times@@@Partition[Fibonacci[Range[30]],4,1] (* Harvey P. Dale, Dec 23 2013 *)
LinearRecurrence[{5,15,-15,-5,1},{6,30,240,1560,10920},30] (* Harvey P. Dale, Jul 24 2021 *)

G.f.: -6*x/((x-1)*(x^2+3*x+1)*(x^2-7*x+1)). - Alois P. Heinz, Oct 02 2013
a(n+5) = 5*a(n+4)+15*a(n+3)-15*a(n+2)-5*a(n+1)+a(n). - Robert Israel, Apr 06 2015
a(n) = 2 * A000217(A059840(n+2)). - Diego Rattaggi, Jan 27 2021
Sum_{n>=1} 1/a(n) = (12-5*sqrt(5))/4. - Diego Rattaggi, Aug 16 2021
a(n) = 3 * Sum_{k=1..n} 2^(n-k)*F(k)^2*F(k+1)*F(k+2) (Seiffert, 2006). - Amiram Eldar, Jan 11 2022

A147779 Squarefree positive integers of the form uv(u^2-v^2) for some integer u,v.

Original entry on oeis.org

6, 30, 210, 330, 546, 2310, 2730, 3570, 4290, 5610, 6090, 6630, 7854, 8970, 9690, 10374, 10626, 13566, 18354, 19866, 22134, 25806, 26970, 39270, 43890, 51330, 51414, 52026, 54834, 56730, 59334, 66990, 68034, 71610, 72930, 74046, 75174

Offset: 1

Max Alekseyev, Nov 12 2008

nonn

Cf. A003273, A006991, A009112, A073120, A147778.

Squarefree terms of A147778. Squarefree terms of A073120. Squarefree terms of A009112.

Terms of A006991 (primitive congruent numbers) corresponding to right triangles with integer sides.

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

cp's OEIS Frontend

A057102 a(n) = 4 * A073120(n).

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A003273 Congruent numbers: positive integers k for which there exists a right triangle having area k and rational sides.

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A009112 Areas of Pythagorean triangles: numbers which can be the area of a right triangle with integer sides.

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A024365 Areas of right triangles with coprime integer sides.

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A256418 Congrua (possible solutions to the congruum problem): numbers k such that there are integers x, y and z with k = x^2-y^2 = z^2-x^2.

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A147778 Positive integers of the form u*v*(u^2 - v^2) where u, v are coprime integers.

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A228873 a(n) = F(n) * F(n+1) * F(n+2) * F(n+3), the product of four consecutive Fibonacci numbers, A000045.

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A147778 Positive integers of the form uv(u^2 - v^2) where u, v are coprime integers.

A147779 Squarefree positive integers of the form uv(u^2-v^2) for some integer u,v.