A269993 -id:A269993 - cp's OEIS frontend

A270349 Denominators of r-Egyptian fraction expansion for sqrt(2) - 1, where r = (1,1/2,1/4,1/8,...)

3, 7, 27, 650, 689392, 1130869248534, 2046949388776880512222550, 5664769376602746621028306587399157369622446276283, 61600875764518391286867927949695082949269716944423018977948114995142883041085134431474743108010213

Offset: 1

Clark Kimberling, Mar 17 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/(2*7) + 1/(4*27) + ...

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, A270347.

r[k_] := 2/2^k; 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) = 2/2^k;
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 18 2016

a(n) = A270347(n+1).

A270350 Denominators of r-Egyptian fraction expansion for sqrt(3) - 1, where r = (1, 1/2, 1/4, 1/8, ...)

Original entry on oeis.org

2, 3, 4, 44, 1446, 3423518, 263631451737996, 70985515555913904515293113895, 8645798497265822420998718966216306501746531100894289290802, 78713180847550502513757221862401308079612732875925186430170968601702893264445327722349352410275677392885249561650440

Offset: 1

Clark Kimberling, Mar 17 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/(2*3) + 1/(4*4) + ...

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_] := 2/2^k; 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) = 2/2^k;
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 18 2016

A270351 Denominators of r-Egyptian fraction expansion for golden ratio - 1, where r = (1, 1/2, 1/4, 1/8, ...)

Original entry on oeis.org

2, 5, 14, 707, 1470654, 1143462781221, 1805535113251940020114035, 2497859054491311040375647235065337168455108737151, 3189945744303964831068292153370103839290925070278698110007359838830245675325591867634500100743606

Offset: 1

Clark Kimberling, Mar 17 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/(2*5) + 1/(4*14) + ...

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_] := 2/2^k; 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; Table[n[x, k], {k, 1, z}]

r(k) = 2/2^k;
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 18 2016

A270352 Denominators of r-Egyptian fraction expansion for 1/Pi, where r = (1, 1/2, 1/4, 1/8, ...)

Original entry on oeis.org

4, 8, 44, 977, 498723, 138012074956, 45087947486104434546449, 2223745971024423874814212532278502253766982404, 3439676840537267257806008796995789895364959784333600339427716437786254731225969490712842205

Offset: 1

Clark Kimberling, Mar 17 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/(2*8) + 1/(4*44) + ...

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_] := 2/2^k; 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) = 2/2^k;
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 18 2016

A270353 Denominators of r-Egyptian fraction expansion for Pi - 3, where r = (1, 1/2, 1/4, 1/8, ...)

Original entry on oeis.org

8, 31, 540, 189864, 22502468823, 547694780221174920178, 287920070745319667821031437298831171428290, 271667810016366767427285213650617821610883263237085072498040538105208873088855853524

Offset: 1

Clark Kimberling, Mar 17 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.

			Pi - 3 = 1/8 + 1/(2*31) + 1/(4*540) + ...

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_] := 2/2^k; 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) = 2/2^k;
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 18 2016

A270354 Denominators of r-Egyptian fraction expansion for 1/e, where r = (1, 1/2, 1/4, 1/8, ...)

Original entry on oeis.org

3, 15, 207, 24777, 1797835772, 2401072239422894903, 36947191921380265723491992928675837908, 1242004943621920150072266455052958650167034792376067355585774287542963919184

Offset: 1

Clark Kimberling, Mar 17 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/(2*15) + 1/(4*207) + ...

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_] := 2/2^k; 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) = 2/2^k;
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 18 2016

A270355 Denominators of r-Egyptian fraction expansion for e - 2, where r = (1, 1/2, 1/4, 1/8, ...)

Original entry on oeis.org

2, 3, 5, 78, 4962, 15925310, 303532967750376, 72884922416996896007616951849, 3238110775186648021853203185875679911508503009261997468560, 7716186732679740909751872277405382774000613384297298421745471878603639986756704754013029661605882827711280194233739

Offset: 1

Clark Kimberling, Mar 17 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 = 1/2 + 1/(2*3) + 1/(4*5) + ...

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_] := 2/2^k; 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) = 2/2^k;
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));

A270356 Denominators of r-Egyptian fraction expansion for log(2), where r = (1, 1/2, 1/4, 1/8, ...)

Original entry on oeis.org

2, 3, 10, 85, 6297, 105324757, 10291333539500676, 72129634294824118806681649563665, 3614136206345221874912341551952565198060297016360952863886217259

Offset: 1

Clark Kimberling, Mar 17 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/(2*3) + 1/(4*10) + ...

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_] := 2/2^k; 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) = 2/2^k;
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 18 2016

A270357 Denominators of r-Egyptian fraction expansion for the Euler-Mascheroni constant, where r = (1, 1/2, 1/4, 1/8, ...)

Original entry on oeis.org

2, 7, 44, 1188, 1107451, 1655310214489, 4507412592442565132297462, 21590918158669845303602195101212593993014272683073, 535939144392644394939678701363249006606218981708849983487820117907080422754959222872984260614611702

Offset: 1

Clark Kimberling, Mar 17 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/(2*7) + 1/(4*44) + ...

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_] := 2/2^k; 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) = 2/2^k;
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 18 2016

A270371 Denominators of r-Egyptian fraction expansion for sqrt(1/2), where r = (1,1/4,1/9,1/16,...).

Original entry on oeis.org

2, 2, 2, 3, 7, 7702, 1234163819, 1590823281229385753, 7255753768720849630767399215373753335, 44436679763085787755205863082559307822924182270889047678247210478391618529

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(1/2) = 1/2 + 1/(4*2) + 1/(9*2) + 1/(16*3) + 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 = Sqrt[1/2]; 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=sqrt(1/2)) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 21 2016

cp's OEIS Frontend

A270349 Denominators of r-Egyptian fraction expansion for sqrt(2) - 1, where r = (1,1/2,1/4,1/8,...)

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A270350 Denominators of r-Egyptian fraction expansion for sqrt(3) - 1, where r = (1, 1/2, 1/4, 1/8, ...)

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A270351 Denominators of r-Egyptian fraction expansion for golden ratio - 1, where r = (1, 1/2, 1/4, 1/8, ...)

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A270352 Denominators of r-Egyptian fraction expansion for 1/Pi, where r = (1, 1/2, 1/4, 1/8, ...)

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A270353 Denominators of r-Egyptian fraction expansion for Pi - 3, where r = (1, 1/2, 1/4, 1/8, ...)

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A270354 Denominators of r-Egyptian fraction expansion for 1/e, where r = (1, 1/2, 1/4, 1/8, ...)

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A270355 Denominators of r-Egyptian fraction expansion for e - 2, where r = (1, 1/2, 1/4, 1/8, ...)

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A270356 Denominators of r-Egyptian fraction expansion for log(2), where r = (1, 1/2, 1/4, 1/8, ...)