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Magata's Constant


MagatasConstantPolys

Consider the Lagrange interpolating polynomial

 f(x)=b_0+(x-1)(b_1+(x-2)(b_3+(x-3)+...))
(1)

through the points (n,p_n), where p_n is the nth prime. For the first few points, the polynomials are

P_1(x)=2
(2)
P_2(x)=1(x-1)+2
(3)
P_3(x)=(1/2(x-2)+1)(x-1)+2
(4)
P_4(x)=((-1/6(x-3)+1/2)(x-2)+1)(x-1)+2
(5)
P_5(x)=(((1/8(x-4)-1/6)(x-3)+1/2)(x-2)+1)×(x-1)+2.
(6)

So the first few values of b_0, b_1, b_2, ..., are 2, 1, 1/2, -1/6, 1/8, -3/40, ... (OEIS A118210 and A118211).

MagatasConstantSums

Now consider the partial sums of these coefficients, namely 2, 3, 7/2, 10/3, 83/24, 203/60, 2459/720, ... (OEIS A118203 and A118204). As first noted by F. Magata in 1998, the sum appears to converge to the value 3.407069... (OEIS A092894), now known as Magata's constant.


See also

Lagrange Interpolating Polynomial

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References

Sloane, N. J. A. Sequences A092894, A118203, A118204, A118210, and A118210 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Magata's Constant

Cite this as:

Weisstein, Eric W. "Magata's Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MagatasConstant.html

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