A269993 -id:A269993 - cp's OEIS frontend

A270372 Denominators of r-Egyptian fraction expansion for sqrt(1/3), where r = (1, 1/4, 1/9, 1/16, ...).

2, 4, 8, 66, 2776, 20101656, 1227318932297655, 8216049453479522437439630860819, 474082010892842884364582298006064172482079224559365990598026496

Offset: 1

Clark Kimberling, Mar 20 2016

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.

See A269993 for a guide to related sequences.

			sqrt(1/3) = 1/2 + 1/(4*4) + 1/(9*2) + 1/(16*3) + 1/(25*7) + ...

Clark Kimberling, Table of n, a(n) for n = 1..12
Eric Weisstein's World of Mathematics, Egyptian Fraction
Index entries for sequences related to Egyptian fractions

Cf. A269993.

r[k_] := 1/k^2; f[x_, 0] = x; z = 10;
n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
x = Sqrt[1/3]; Table[n[x, k], {k, 1, z}]

r(k) = 1/k^2;
f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
a(k, x=sqrt(1/3)) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 21 2016

A270373 Denominators of r-Egyptian fraction expansion for sqrt(2) - 1, where r = (1, 1/4, 1/9, 1/16, ...).

Original entry on oeis.org

3, 4, 7, 25, 5546, 36482088, 14423934280776257, 1969937215073991451613042447271867, 3160555685801520768089757205744771458914199650397475324265981061618

Offset: 1

Clark Kimberling, Mar 20 2016

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.

See A269993 for a guide to related sequences.

			sqrt(2) - 1 = 1/3 + 1/(4*4) + 1/(9*7) + 1/(16*25) + ...

Clark Kimberling, Table of n, a(n) for n = 1..12
Eric Weisstein's World of Mathematics, Egyptian Fraction
Index entries for sequences related to Egyptian fractions

Cf. A269993.

r[k_] := 1/k^2; f[x_, 0] = x; z = 10;
n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
x = Sqrt[2] - 1; Table[n[x, k], {k, 1, z}]

r(k) = 1/k^2;
f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
a(k, x=sqrt(2)-1) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 21 2016

A270374 Denominators of r-Egyptian fraction expansion for sqrt(3) - 1, where r = (1,1/4,1/9,1/16,...).

Original entry on oeis.org

2, 2, 2, 2, 2, 114, 12858, 155940365, 49973147636187261, 2858100604081391412323339697785029, 9144547702051996958048744386280174102458208170798737295487273148214

Offset: 1

Clark Kimberling, Mar 20 2016

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.

See A269993 for a guide to related sequences.

			sqrt(3) - 1 = 1/2 + 1/(4*2) + 1/(9*2) + 1/(16*2) + ...

Clark Kimberling, Table of n, a(n) for n = 1..13
Eric Weisstein's World of Mathematics, Egyptian Fraction
Index entries for sequences related to Egyptian fractions

Cf. A269993.

r[k_] := 1/k^2; f[x_, 0] = x; z = 10;
n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
x = Sqrt[3] - 1; Table[n[x, k], {k, 1, z}]

r(k) = 1/k^2;
f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
a(k, x=sqrt(3)-1) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 21 2016

A270375 Denominators of r-Egyptian fraction expansion for golden ratio - 1, where r = (1,1/4,1/9,1/16,...).

Original entry on oeis.org

2, 3, 4, 10, 60, 4473, 23403582, 1295226544484567, 9611349042287513051537445592891, 89998772942534105602452834114784063917358549011796155052807149

Offset: 1

Clark Kimberling, Mar 20 2016

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.

See A269993 for a guide to related sequences.

			tau - 1 = 1/2 + 1/(4*3) + 1/(9*4) + 1/(16*10) + ...

Clark Kimberling, Table of n, a(n) for n = 1..13
Eric Weisstein's World of Mathematics, Egyptian Fraction
Index entries for sequences related to Egyptian fractions

Cf. A269993.

r[k_] := 1/k^2; f[x_, 0] = x; z = 10;
n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
x = GoldenRatio - 1; Table[n[x, k], {k, 1, z}]

r(k) = 1/k^2;
f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
a(k, x=(sqrt(5)-1)/2) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 21 2016

A270376 Denominators of r-Egyptian fraction expansion for 1/Pi, where r = (1, 1/4, 1/9, 1/16, ...).

Original entry on oeis.org

4, 4, 20, 246, 150610, 28628772458, 4633718454684972107216, 32270052939985266099596531363945117655631355, 1531822012919710742180024988940181184501391371231597927081244972822399811859680584475111

Offset: 1

Clark Kimberling, Mar 20 2016

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.

See A269993 for a guide to related sequences.

			1/Pi = 1/4 + 1/(4*4) + 1/(9*20) + 1/(16*246) + ...

Clark Kimberling, Table of n, a(n) for n = 1..12
Eric Weisstein's World of Mathematics, Egyptian Fraction
Index entries for sequences related to Egyptian fractions

Cf. A269993.

r[k_] := 1/k^2; f[x_, 0] = x; z = 10;
n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
x = 1/Pi; Table[n[x, k], {k, 1, z}]

r(k) = 1/k^2;
f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
a(k, x=1/Pi) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 21 2016

A270377 Denominators of r-Egyptian fraction expansion for Pi - 3, where r = (1,1/4,1/9,1/16,...).

Original entry on oeis.org

8, 16, 115, 42517, 2725016283, 22037592325978294230, 376949052509622237440534036730873293477, 162105898616252691011784334305248213903014362390225130418238883927812046205359

Offset: 1

Clark Kimberling, Mar 20 2016

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.

See A269993 for a guide to related sequences.

			1/Pi = 1/8 + 1/(4*16) + 1/(9*115) + 1/(16*42517) + ...

Clark Kimberling, Table of n, a(n) for n = 1..10
Eric Weisstein's World of Mathematics, Egyptian Fraction
Index entries for sequences related to Egyptian fractions

Cf. A269993.

r[k_] := 1/k^2; f[x_, 0] = x; z = 10;
n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
x = Pi - 3; Table[n[x, k], {k, 1, z}]

r(k) = 1/k^2;
f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
a(k, x=Pi-3) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 21 2016

A270378 Denominators of r-Egyptian fraction expansion for 1/e, where r = (1, 1/4, 1/9, 1/16, ...).

Original entry on oeis.org

3, 8, 34, 2222, 6483909, 53731622976437, 3099497943165662781193803037, 63757313155180253672051718522425349303280644466076099608, 44373380497244637497779460270147771148709175688739800767598157179085876588140068013506306978166146857743130359405

Offset: 1

Clark Kimberling, Mar 20 2016

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.

See A269993 for a guide to related sequences.

			1/e = 1/3 + 1/(4*8) + 1/(9*34) + 1/(16*2222) + ...

Clark Kimberling, Table of n, a(n) for n = 1..11
Eric Weisstein's World of Mathematics, Egyptian Fraction
Index entries for sequences related to Egyptian fractions

Cf. A269993.

r[k_] := 1/k^2; f[x_, 0] = x; z = 10;
n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
x = 1/E; Table[n[x, k], {k, 1, z}]

r(k) = 1/k^2;
f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
a(k, x=exp(-1)) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 21 2016

A270379 Denominators of r-Egyptian fraction expansion for e - 2, where r = (1,1/4,1/9,1/16,...).

Original entry on oeis.org

2, 2, 2, 2, 7, 37, 1817, 3361666, 24283670558553, 1002770956493811911694552768, 843337841302004296404319706946194895734287215696998151, 890614335579920230119707369559263943501588363957602897846451247124061017888881480680329044310526970935480485

Offset: 1

Clark Kimberling, Mar 20 2016

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.

See A269993 for a guide to related sequences.

			e - 2 = 1/2 + 1/(2*2) + 1/(9*2) + 1/(16*2) + 1/(25*7) + ...

Clark Kimberling, Table of n, a(n) for n = 1..13
Eric Weisstein's World of Mathematics, Egyptian Fraction
Index entries for sequences related to Egyptian fractions

Cf. A269993.

r[k_] := 1/k^2; f[x_, 0] = x; z = 10;
n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
x = E - 2; Table[n[x, k], {k, 1, z}]

r(k) = 1/k^2;
f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
a(k, x=exp(1)-2) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 21 2016

A270380 Denominators of r-Egyptian fraction expansion for log(2), where r = (1,1/4,1/9,1/16,...).

Original entry on oeis.org

2, 2, 2, 5, 437, 302513, 105578997456, 43374115871914236473230, 3061387825626545156370895519674610145110416238, 7629373903475555167995753649986186306421860702154239431571840759012861049320099293062744031

Offset: 1

Clark Kimberling, Mar 20 2016

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.

See A269993 for a guide to related sequences.

			log(2) = 1/2 + 1/(4*2) + 1/(9*2) + 1/(16*5) + 1/(25*437) + ...

Clark Kimberling, Table of n, a(n) for n = 1..13
Eric Weisstein's World of Mathematics, Egyptian Fraction
Index entries for sequences related to Egyptian fractions

Cf. A269993.

r[k_] := 1/k^2; f[x_, 0] = x; z = 10;
n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
x = Log(2); Table[n[x, k], {k, 1, z}]

r(k) = 1/k^2;
f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
a(k, x=log(2)) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 21 2016

A270381 Denominators of r-Egyptian fraction expansion for the Euler-Mascheroni constant (EulerGamma), where r = (1, 1/4, 1/9, 1/16, ...).

Original entry on oeis.org

2, 4, 8, 76, 9076, 76367616, 8970505418837847, 105072872079226455591148215294656, 26036426776932682309708313847332475352524205664479930813102637500

Offset: 1

Clark Kimberling, Mar 20 2016

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.

See A269993 for a guide to related sequences.

			Euler-Mascheroni constant = 1/2 + 1/(4*4) + 1/(9*8) + 1/(16*76) + ...

Clark Kimberling, Table of n, a(n) for n = 1..12
Eric Weisstein's World of Mathematics, Egyptian Fraction
Index entries for sequences related to Egyptian fractions

Cf. A269993.

r[k_] := 1/k^2; f[x_, 0] = x; z = 10;
n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
x = EulerGamma; Table[n[x, k], {k, 1, z}]

r(k) = 1/k^2;
f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
a(k, x=Euler) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 21 2016

cp's OEIS Frontend

A270372 Denominators of r-Egyptian fraction expansion for sqrt(1/3), where r = (1, 1/4, 1/9, 1/16, ...).

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A270373 Denominators of r-Egyptian fraction expansion for sqrt(2) - 1, where r = (1, 1/4, 1/9, 1/16, ...).

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A270374 Denominators of r-Egyptian fraction expansion for sqrt(3) - 1, where r = (1,1/4,1/9,1/16,...).

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A270375 Denominators of r-Egyptian fraction expansion for golden ratio - 1, where r = (1,1/4,1/9,1/16,...).

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A270376 Denominators of r-Egyptian fraction expansion for 1/Pi, where r = (1, 1/4, 1/9, 1/16, ...).

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A270377 Denominators of r-Egyptian fraction expansion for Pi - 3, where r = (1,1/4,1/9,1/16,...).

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A270378 Denominators of r-Egyptian fraction expansion for 1/e, where r = (1, 1/4, 1/9, 1/16, ...).

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A270379 Denominators of r-Egyptian fraction expansion for e - 2, where r = (1,1/4,1/9,1/16,...).

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