A020760 -id:A020760 - cp's OEIS frontend

A372273 Decimal expansion of the largest positive zero of the Legendre polynomial of degree 6.

9, 3, 2, 4, 6, 9, 5, 1, 4, 2, 0, 3, 1, 5, 2, 0, 2, 7, 8, 1, 2, 3, 0, 1, 5, 5, 4, 4, 9, 3, 9, 9, 4, 6, 0, 9, 1, 3, 4, 7, 6, 5, 7, 3, 7, 7, 1, 2, 2, 8, 9, 8, 2, 4, 8, 7, 2, 5, 4, 9, 6, 1, 6, 5, 2, 6, 6, 1, 3, 5, 0, 0, 8, 4, 4, 2, 0, 0, 1, 9, 6, 2, 7, 6, 2, 8, 8

Offset: 0

Pontus von Brömssen, Apr 25 2024

			0.932469514203152027812301554493994609134765737712289824872549...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Wikipedia, Legendre polynomials.
Index entries for algebraic numbers, degree 6.

Cf. A008316, A100258.
There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
   k | zeros
  ---+--------------------------
   2 | A020760
   3 | A010513/10
   4 | A372267, A372268
   5 | A372269, A372270
   6 | A372271, A372272, A372273
   7 | A372274, A372275, A372276

First[RealDigits[Root[LegendreP[6, #] &, 6], 10, 100]] (* Paolo Xausa, Feb 27 2025 *)

Largest positive root of 231*x^6 - 315*x^4 + 105*x^2 - 5 = 0.

A372274 Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 7.

Original entry on oeis.org

4, 0, 5, 8, 4, 5, 1, 5, 1, 3, 7, 7, 3, 9, 7, 1, 6, 6, 9, 0, 6, 6, 0, 6, 4, 1, 2, 0, 7, 6, 9, 6, 1, 4, 6, 3, 3, 4, 7, 3, 8, 2, 0, 1, 4, 0, 9, 9, 3, 7, 0, 1, 2, 6, 3, 8, 7, 0, 4, 3, 2, 5, 1, 7, 9, 4, 6, 6, 3, 8, 1, 3, 2, 2, 6, 1, 2, 5, 6, 5, 5, 3, 2, 8, 3, 1, 2

Offset: 0

Pontus von Brömssen, Apr 25 2024

			0.405845151377397166906606412076961463347382014099370126387043...

Paolo Xausa, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Table 25.4, n=7.
A.H.M. Smeets, Python program for Legendre-Gauss quadrature constants.
Eric Weisstein's World of Mathematics, Legendre Polynomial.
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature.
Wikipedia, Legendre polynomials.
Index entries for algebraic numbers, degree 6.

Cf. A008316, A100258.
There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
   k | zeros                           | corresponding weights for Legendre-Gauss quadrature
  ---+---------------------------------+----------------------------------------------------
   2 | A020760                         | A000007*10
   3 | A010513/10                      | A010716
   4 | A372267, A372268                | A382103, A382104
   5 | A372269, A372270                | A382105, A382106
   6 | A372271, A372272, A372273       | A382107, A382686, A382687
   7 | this sequence, A372275, A372276 | A382688, A382689, A382690

First[RealDigits[Root[LegendreP[7, #] &, 5], 10, 100]] (* Paolo Xausa, Feb 27 2025 *)

solve (x = 0.1, 0.5, 429*x^6 - 693*x^4 + 315*x^2 - 35) \\ A.H.M. Smeets, May 31 2025

Smallest positive root of 429*x^6 - 693*x^4 + 315*x^2 - 35 = 0.

A372275 Decimal expansion of the middle positive zero of the Legendre polynomial of degree 7.

Original entry on oeis.org

7, 4, 1, 5, 3, 1, 1, 8, 5, 5, 9, 9, 3, 9, 4, 4, 3, 9, 8, 6, 3, 8, 6, 4, 7, 7, 3, 2, 8, 0, 7, 8, 8, 4, 0, 7, 0, 7, 4, 1, 4, 7, 6, 4, 7, 1, 4, 1, 3, 9, 0, 2, 6, 0, 1, 1, 9, 9, 5, 5, 3, 5, 1, 9, 6, 7, 4, 2, 9, 8, 7, 4, 6, 7, 2, 1, 8, 0, 5, 1, 3, 7, 9, 2, 8, 2, 6

Offset: 0

Pontus von Brömssen, Apr 25 2024

			0.741531185599394439863864773280788407074147647141390260119955...

Paolo Xausa, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Table 25.4, n=7.
A.H.M. Smeets, Python program for Legendre-Gauss quadrature constants.
Eric Weisstein's World of Mathematics, Legendre Polynomial.
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature.
Wikipedia, Legendre polynomials.
Index entries for algebraic numbers, degree 6.

Cf. A008316, A100258.
There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
   k | zeros                           | corresponding weights for Legendre-Gauss quadrature
  ---+---------------------------------+----------------------------------------------------
   2 | A020760                         | A000007*10
   3 | A010513/10                      | A010716
   4 | A372267, A372268                | A382103, A382104
   5 | A372269, A372270                | A382105, A382106
   6 | A372271, A372272, A372273       | A382107, A382686, A382687
   7 | A372274, this sequence, A372276 | A382688, A382689, A382690

First[RealDigits[Root[LegendreP[7, #] &, 6], 10, 100]] (* Paolo Xausa, Feb 27 2025 *)

solve (x = 0.6, 0.8, 429*x^6 - 693*x^4 + 315*x^2 - 35) \\ A.H.M. Smeets, May 31 2025

Middle positive root of 429*x^6 - 693*x^4 + 315*x^2 - 35 = 0.

A372276 Decimal expansion of the largest positive zero of the Legendre polynomial of degree 7.

Original entry on oeis.org

9, 4, 9, 1, 0, 7, 9, 1, 2, 3, 4, 2, 7, 5, 8, 5, 2, 4, 5, 2, 6, 1, 8, 9, 6, 8, 4, 0, 4, 7, 8, 5, 1, 2, 6, 2, 4, 0, 0, 7, 7, 0, 9, 3, 7, 6, 7, 0, 6, 1, 7, 7, 8, 3, 5, 4, 8, 7, 6, 9, 1, 0, 3, 9, 1, 3, 0, 6, 3, 3, 3, 0, 3, 5, 4, 8, 4, 0, 1, 4, 0, 8, 0, 5, 7, 3, 0

Offset: 0

Pontus von Brömssen, Apr 25 2024

			0.949107912342758524526189684047851262400770937670617783548769...

Paolo Xausa, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Table 25.4, n=7.
A.H.M. Smeets, Python program for Legendre-Gauss quadrature constants.
Eric Weisstein's World of Mathematics, Legendre Polynomial.
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature.
Wikipedia, Legendre polynomials.
Index entries for algebraic numbers, degree 6.

Cf. A008316, A100258.
There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
   k | zeros                           | corresponding weights for Legendre-Gauss quadrature
  ---+---------------------------------+----------------------------------------------------
   2 | A020760                         | A000007*10
   3 | A010513/10                      | A010716
   4 | A372267, A372268                | A382103, A382104
   5 | A372269, A372270                | A382105, A382106
   6 | A372271, A372272, A372273       | A382107, A382686, A382687
   7 | A372274, A372275, this sequence | A382688, A382689, A382690

First[RealDigits[Root[LegendreP[7, #] &, 7], 10, 100]] (* Paolo Xausa, Feb 27 2025 *)

solve (x = 0.8, 1.0, 429*x^6 - 693*x^4 + 315*x^ - 35) \\ A.H.M. Smeets, May 31 2025

Largest positive root of 429*x^6 - 693*x^4 + 315*x^2 - 35 = 0.

A020765 Decimal expansion of 1/sqrt(8).

Original entry on oeis.org

3, 5, 3, 5, 5, 3, 3, 9, 0, 5, 9, 3, 2, 7, 3, 7, 6, 2, 2, 0, 0, 4, 2, 2, 1, 8, 1, 0, 5, 2, 4, 2, 4, 5, 1, 9, 6, 4, 2, 4, 1, 7, 9, 6, 8, 8, 4, 4, 2, 3, 7, 0, 1, 8, 2, 9, 4, 1, 6, 9, 9, 3, 4, 4, 9, 7, 6, 8, 3, 1, 1, 9, 6, 1, 5, 5, 2, 6, 7, 5, 9, 7, 1, 2, 5, 9, 6, 8, 8, 3, 5, 8, 1, 9, 1, 0, 3, 9, 3

Offset: 0

N. J. A. Sloane

Multiplied by 10, this is the real and the imaginary part of sqrt(25i). - Alonso del Arte, Jan 11 2013
Radius of the midsphere (tangent to the edges) in a regular tetrahedron with unit edges. - Stanislav Sykora, Nov 20 2013
The side of the largest cubical present that can be wrapped (with cutting) by a unit square of wrapping paper. See Problem 10716 link. - Michel Marcus, Jul 24 2018
The ratio between the thickness and diameter of a geometrically fair coin having an equal probability, 1/3, of landing on each of its two faces and on its side after being tossed in the air. The calculation is based on comparing the areal projections of the faces and sides of the coin on a circumscribing sphere. (Mosteller, 1965). See A020760 for a physical solution. - Amiram Eldar, Sep 01 2020

			1/sqrt(8) = 0.353553390593273762200422181052424519642417968844237018294...

Frederick Mosteller, Fifty challenging problems of probability, Dover, New York, 1965. See problem 38, pp. 10 and 58-60.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Michael L. Catalano-Johnson, Daniel Loeb and John Beebee, A cubical gift: Problem 10716, The American Mathematical Monthly, Vol. 108, No. 1 (Jan., 2001), pp. 81-82.
Wikipedia, Tetrahedron.
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 2

Cf. Midsphere radii in Platonic solids:
A020761 (octahedron),
A010503 (cube),
A019863 (icosahedron),
A239798 (dodecahedron).

Digits:=100; evalf(1/sqrt(8)); # Wesley Ivan Hurt, Mar 27 2014

RealDigits[N[1/Sqrt[8], 200]] (* Vladimir Joseph Stephan Orlovsky, May 27 2010 *)
realDigitsRecip[Sqrt[8]] (* The realDigitsRecip program is at A021200 *) (* Harvey P. Dale, Apr 05 2025 *)

sqrt(1/8) \\ Charles R Greathouse IV, Apr 25 2016

A010503 divided by 2.
Equals A201488 minus 1/2. Equals 1/(A010487-4) minus 1/4. - Jon E. Schoenfield, Jan 09 2017
Equals Integral_{x=0..oo} x*exp(-x)*BesselJ(0,x) dx. - Kritsada Moomuang, Jun 03 2025

A050931 Numbers having a prime factor congruent to 1 mod 6.

Original entry on oeis.org

7, 13, 14, 19, 21, 26, 28, 31, 35, 37, 38, 39, 42, 43, 49, 52, 56, 57, 61, 62, 63, 65, 67, 70, 73, 74, 76, 77, 78, 79, 84, 86, 91, 93, 95, 97, 98, 103, 104, 105, 109, 111, 112, 114, 117, 119, 122, 124, 126, 127, 129, 130, 133, 134, 139, 140, 143, 146, 147, 148, 151

Offset: 1

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 30 1999

Original definition: Solutions c of cot(2*Pi/3)*(-(a+b+c)*(-a+b+c)*(-a+b-c)*(a+b-c))^(1/2)=a^2+b^2-c^2, c>a,b integers.
Note cot(2*Pi/3) = -1/sqrt(3).
Also the c-values for solutions to c^2 = a^2 + ab + b^2 in positive integers. Also the numbers which occur as the longest side of some triangle with integer sides and a 120-degree angle. - Paul Boddington, Nov 05 2007
The sequence can also be defined as the numbers w which are Heronian means of two distinct positive integers u and v, i.e., w = [u+sqrt(uv)+v]/3. E.g., 28 is the Heronian mean of 4 and 64 (and also of 12 and 48). - Pahikkala Jussi, Feb 16 2008
From Jean-Christophe Hervé, Nov 24 2013: (Start)
This sequence is the analog of hypotenuse numbers A009003 for triangles with integer sides and a 120-degree angle. There are two integers a and b > 0 such that a(n)^2 = a^2 + ab + b^2, and a, b and a(n) are the sides of the triangle: a(n) is the sequence of lengths of the longest side of these triangles. A004611 is the same for primitive triangles.
a and b cannot be equal because sqrt(3) is not rational. Then the values a(n) are such that a(n)^2 is in A024606. It follows that a(n) is the sequence of multiples of primes of form 6k+1 A002476.
The sequence is closed under multiplication. The primitive elements are those with exactly one prime divisor of the form 6k+1 with multiplicity one, which are also those for which there exists a unique 120-degree integer triangle with its longest side equals to a(n).
(End)
Conjecture: Numbers m such that abs(Sum_{k=1..m} [k|m]*A008683(k)*(-1)^(2*k/3)) = 0. - Mats Granvik, Jul 06 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Bojan Mohar, Hermitian adjacency spectrum and switching equivalence of mixed graphs, arXiv preprint arXiv:1505.03373 [math.CO], 2015.
Planet Math, Truncated cone
Eric Weisstein's World of Mathematics, Triangle - see especially (19)
Index entries for sequences related to A2 = hexagonal = triangular lattice

Cf. A002476, A004611, A024606, A230780 (complement), A009003.

Cf. A027748.

a050931 n = a050931_list !! (n-1)
a050931_list = filter (any (== 1) . map (flip mod 6) . a027748_row) [1..]
-- Reinhard Zumkeller, Apr 09 2014

Select[Range[2,200],MemberQ[Union[Mod[#,6]&/@FactorInteger[#][[All,1]]],1]&] (* Harvey P. Dale, Aug 24 2019 *)

is_A050931(n)=n>6&&Set(factor(n)[,1]%6)[1]==1 \\ M. F. Hasler, Mar 04 2018

A005088(a(n)) > 0. Terms are obtained by the products A230780(k)*A004611(p) for k, p > 0, ordered by increasing values. - Jean-Christophe Hervé, Nov 24 2013

cot(2*Pi/3) = -1/sqrt(3) = -0.57735... = - A020760. - M. F. Hasler, Aug 18 2016

Simpler definition from M. F. Hasler, Mar 04 2018

A382103 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372267.

Original entry on oeis.org

3, 4, 7, 8, 5, 4, 8, 4, 5, 1, 3, 7, 4, 5, 3, 8, 5, 7, 3, 7, 3, 0, 6, 3, 9, 4, 9, 2, 2, 1, 9, 9, 9, 4, 0, 7, 2, 3, 5, 3, 4, 8, 6, 9, 5, 8, 3, 3, 8, 9, 3, 5, 4, 0, 4, 9, 2, 5, 2, 9, 3, 1, 9, 5, 1, 8, 7, 5, 1, 8, 6, 7, 4, 6, 5, 9, 1, 0, 3, 5, 1, 7, 2, 1, 9, 8, 3

Offset: 0

A.H.M. Smeets, Mar 15 2025

There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
   k | zeros                     | corresponding weights for Legendre-Gauss quadrature
  ---+---------------------------+----------------------------------------------------
   2 | A020760                   | A000007*10
   3 | A010513/10                | A010716
   4 | A372267, A372268          | this sequence, A382104
   5 | A372269, A372270          | A382105, A382106
   6 | A372271, A372272, A372273 | A382107, A382686, A382687
   7 | A372274, A372275, A372276 | A382688, A382689, A382690

			0.34785484513745385737306394922199940723534869583389...

A.H.M. Smeets, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Table 25.4, n=4.
A.H.M. Smeets, Python program for Legendre-Gauss quadrature constants.
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature.

Cf. A372267.

RealDigits[1/2 - Sqrt[5/6]/6, 10, 120][[1]] (* Amiram Eldar, Mar 24 2025 *)

Equals 1/2 - (1/6)*sqrt(5/6).

A382104 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372268.

Original entry on oeis.org

6, 5, 2, 1, 4, 5, 1, 5, 4, 8, 6, 2, 5, 4, 6, 1, 4, 2, 6, 2, 6, 9, 3, 6, 0, 5, 0, 7, 7, 8, 0, 0, 0, 5, 9, 2, 7, 6, 4, 6, 5, 1, 3, 0, 4, 1, 6, 6, 1, 0, 6, 4, 5, 9, 5, 0, 7, 4, 7, 0, 6, 8, 0, 4, 8, 1, 2, 4, 8, 1, 3, 2, 5, 3, 4, 0, 8, 9, 6, 4, 8, 2, 7, 8, 0, 1, 6

Offset: 0

A.H.M. Smeets, Mar 15 2025

There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
   k | zeros                     | corresponding weights for Legendre-Gauss quadrature
  ---+---------------------------+----------------------------------------------------
   2 | A020760                   | A000007*10
   3 | A010513/10                | A010716
   4 | A372267, A372268          | A382103, this sequence
   5 | A372269, A372270          | A382105, A382106
   6 | A372271, A372272, A372273 | A382107, A382686, A382687
   7 | A372274, A372275, A372276 | A382688, A382689, A382690

			0.65214515486254614262693605077800059276465130416610645...

A.H.M. Smeets, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Table 25.4, n=4.
A.H.M. Smeets, Python program for Legendre-Gauss quadrature constants.
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature.

Cf. A372268.

RealDigits[1/2 + Sqrt[5/6]/6, 10, 120][[1]] (* Amiram Eldar, Mar 24 2025 *)

1/2 + (1/6)*sqrt(5/6) \\ Stefano Spezia, May 22 2025

Equals 1/2 + (1/6)*sqrt(5/6).

Minimal polynomial: 216*x^2 - 216*x + 49. - Stefano Spezia, May 22 2025

A382106 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372270.

Original entry on oeis.org

2, 3, 6, 9, 2, 6, 8, 8, 5, 0, 5, 6, 1, 8, 9, 0, 8, 7, 5, 1, 4, 2, 6, 4, 0, 4, 0, 7, 1, 9, 9, 1, 7, 3, 6, 2, 6, 4, 3, 2, 6, 0, 0, 0, 2, 2, 1, 2, 4, 1, 4, 0, 1, 5, 5, 8, 2, 8, 2, 7, 8, 8, 8, 2, 2, 1, 7, 1, 7, 2, 8, 8, 4, 0, 3, 0, 4, 3, 0, 9, 8, 5, 7, 9, 9, 9, 3

Offset: 0

A.H.M. Smeets, Mar 27 2025

There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
 k | zeros                     | corresponding weights
---+---------------------------+--------------------------
 2 | A020760                   | A000007*10
 3 | A010513/10                | A010716
 4 | A372267, A372268          | A382103, A382104
 5 | A372269, A372270          | A382106, this sequence
 6 | A372271, A372272, A372273 | A382107, A382686, A382687
 7 | A372274, A372275, A372276 | A382688, A382689, A382690

			0.236926885056189087514264040719917362643260002212...

A.H.M. Smeets, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Table 25.4, n=5.
A.H.M. Smeets, Python program for Legendre-Gauss quadrature constants.
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature.

Cf. A372270.

Equals (322-13*sqrt(70))/900.

A382107 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372271.

Original entry on oeis.org

4, 6, 7, 9, 1, 3, 9, 3, 4, 5, 7, 2, 6, 9, 1, 0, 4, 7, 3, 8, 9, 8, 7, 0, 3, 4, 3, 9, 8, 9, 5, 5, 0, 9, 9, 4, 8, 1, 1, 6, 5, 5, 6, 0, 5, 7, 6, 9, 2, 1, 0, 5, 3, 5, 3, 1, 1, 6, 2, 5, 3, 1, 9, 9, 6, 3, 9, 1, 4, 2, 0, 1, 6, 2, 0, 3, 9, 8, 1, 2, 7, 0, 3, 1, 1, 1, 0

Offset: 0

A.H.M. Smeets, Mar 27 2025

There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
 k | zeros                     | corresponding weights
---+---------------------------+--------------------------
 2 | A020760                   | A000007*10
 3 | A010513/10                | A010716
 4 | A372267, A372268          | A382103, A382104
 5 | A372269, A372270          | A382105, A382106
 6 | A372271, A372272, A372273 | this sequence, A382686, A382687
 7 | A372274, A372275, A372276 | A382688, A382689, A382690

			0.4679139345726910473898703439895509948116556057692...

A.H.M. Smeets, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Table 25.4, n=6.
A.H.M. Smeets, Python program for Legendre-Gauss quadrature constants.
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature.

Cf. A372271.

cp's OEIS Frontend

A372273 Decimal expansion of the largest positive zero of the Legendre polynomial of degree 6.

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A372274 Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 7.

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A372275 Decimal expansion of the middle positive zero of the Legendre polynomial of degree 7.

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A372276 Decimal expansion of the largest positive zero of the Legendre polynomial of degree 7.

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A020765 Decimal expansion of 1/sqrt(8).

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A050931 Numbers having a prime factor congruent to 1 mod 6.

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A382103 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372267.

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