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A270486 Denominators of r-Egyptian fraction expansion for the Euler-Mascheroni constant (EulerGamma), where r(k) = 1/Prime(k). 1

%I

%S 1,5,19,6299,29244983,2480906174586499,

%T 9602583972368431818444689851565,

%U 310901734358858530002531740085708821780321504952699749074732105

%N Denominators of r-Egyptian fraction expansion for the Euler-Mascheroni constant (EulerGamma), where r(k) = 1/Prime(k).

%C Suppose that r is a sequence of rational numbers r(k) <= 1 for k >= 1, and that x is an irrational number in (0,1). Let f(0) = x, n(k) = floor(r(k)/f(k-1)), and f(k) = f(k-1) - r(k)/n(k). Then x = r(1)/n(1) + r(2)/n(2) + r(3)/n(3) + ... , the r-Egyptian fraction for x.

%C See A269993 for a guide to related sequences.

%H Clark Kimberling, <a href="/A270486/b270486.txt">Table of n, a(n) for n = 1..11</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>

%H <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>

%e Euler-Mascheroni constant = 1/(2*1) + 1/(3*5) + 1/(5*19) + 1/(7*6299) + ...

%t r[k_] := 1/Prime[k]; f[x_, 0] = x; z = 10;

%t n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]

%t f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]

%t x = EulerGamma; Table[n[x, k], {k, 1, z}]

%o (PARI) r(k) = 1/prime(k);

%o f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););

%o a(k, x=Euler) = ceil(r(k)/f(k-1, x)); \\ _Michel Marcus_, Mar 31 2016

%Y Cf. A269993, A000040.

%K nonn,frac,easy

%O 1,2

%A _Clark Kimberling_, Mar 30 2016

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Last modified May 14 13:20 EDT 2021. Contains 343884 sequences. (Running on oeis4.)