A269993 -id:A269993 - cp's OEIS frontend

A269999 Denominators of r-Egyptian fraction expansion for Pi - 3, where r = (1,1/2,1/3,1/4,...)

8, 31, 719, 17276711, 557951558165893, 1713250424923433306065171045669, 3960162768997467999491098138568123635738830147395528618636887, 148114266323338300606167235125265318767829304330791212171374192569332869541220746054882408155611146661783688512870116687748

Offset: 1

Clark Kimberling, Mar 15 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/(3*719) + ...

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; 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;
x = Pi - 3;
f(x, k) = if(k<1, x, f(x, k - 1) - r(k)/n(x, k));
n(x, k) = ceil(r(k)/f(x, k - 1));
for(k = 1, 8, print1(n(x, k), ", ")) \\ Indranil Ghosh, Mar 29 2017

A270001 Denominators of r-Egyptian fraction expansion for 1/e, where r = (1,1/2,1/3,1/4,...)

Original entry on oeis.org

3, 15, 275, 382677, 1046649251798, 2422932913436254796909358, 7298956212857760367589586285406004970615840077289, 146254918268677439622519920044753993861333975570554456887622664610038423045957554361330873143236585

Offset: 1

Clark Kimberling, Mar 15 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/(3*275) + ...

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; 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}]

A270002 Denominators of r-Egyptian fraction expansion for e - 2, where r = (1,1/2,1/3,1/4,...)

Original entry on oeis.org

2, 3, 7, 63, 7179, 142233093, 64600110035609517, 5529148350206824361693538422450743, 39876890198849678230595649918157265458164953427845442505533508344048

Offset: 1

Clark Kimberling, Mar 15 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*3) + 1/(3*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; 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}]

A270314 Denominators of r-Egyptian fraction expansion for log(2), where r = (1,1/2,1/3,1/4,...).

Original entry on oeis.org

2, 3, 13, 298, 355823, 306479173303, 85372761970827958806466, 16575976283809775714654644103484953548013865676, 269025959411335919672976939610798847100114463059537709191005089031919232139117472577538965440

Offset: 1

Clark Kimberling, Mar 15 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/(3*13) + ...

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; 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}]

A270315 Denominators of r-Egyptian fraction expansion for the Euler-Mascheroni constant (Gamma-Euler constant), where r = (1,1/2,1/3,1/4,...)

Original entry on oeis.org

2, 7, 58, 6256, 37041488, 3283456941510566, 87990824525320083189557345568930, 6787481189341615675664690311149906782682845820114751821172918190

Offset: 1

Clark Kimberling, Mar 15 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*7) + 1/(3*58) + ...

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; 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 = GammaEuler; Table[n[x, k], {k, 1, z}]

A270316 Denominators of r-Egyptian fraction expansion for (1/2)^(1/3), where r = (1,1/2,1/3,1/4,...)

Original entry on oeis.org

2, 2, 8, 123, 149367, 19877572990, 3398650153657920854371, 38501744904404393452660892011327652171148221, 1751742507912624184333715455628345093210972368514121272905550101268413741408122585972087

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.

			(1/2)^(1/3) = 1/2 + 1/(2*2) + 1/(3*8) + ...

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; 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 = (1/2)^(1/3); Table[n[x, k], {k, 1, z}]

A270358 Denominators of r-Egyptian fraction expansion for (1/2)^(1/3), where r = (1, 1/2, 1/4, 1/8, ...).

Original entry on oeis.org

2, 2, 6, 62, 3526, 6487141, 39385964848219, 870200535339836766981506923, 7107112253865886739857942326428066600374758700504057908, 51149853017945104127158581151674618357470586573041429321297826264898103722100928190358789489996748918377200334

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.

			(1/2)^(1/3) = 1/2 + 1/(2*2) + 1/(4*6) + ...

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 = (1/2)^(1/3); Table[n[x, k], {k, 1, z}]

A270394 Denominators of r-Egyptian fraction expansion for sqrt(1/2), where r(k) = 1/Fibonacci(k+1).

Original entry on oeis.org

2, 3, 9, 59, 9437, 62059971, 2813586350787717, 8534689167911295735140758101600, 54171527001975050997893888972139886506909953999125751170768531

Offset: 1

Clark Kimberling, Mar 22 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/(2*3) + 1/(3*9) + 1/(5*59) + ...

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, A000045, A010503.

r[k_] := 1/Fibonacci[k+1]; 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/fibonacci(k+1);
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 22 2016

A270405 Denominators of r-Egyptian fraction expansion for (1/2)^(1/3), where r(k) = 1/Fibonacci(k+1).

Original entry on oeis.org

2, 2, 8, 99, 9153, 134325943, 17980902816814494, 336913028495678415812394391065577, 70730509948452535771375914216285436007372776802180962851035180747

Offset: 1

Clark Kimberling, Mar 30 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.

			(1/2)^(1/3) = 1/2 + 1/(2*2) + 1/(3*8) + 1/(5*99) + ...

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, A000045, A270714.

r[k_] := 1/Fibonacci[k+1]; 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 = (1/2)^(1/3); Table[n[x, k], {k, 1, z}]

A270476 Denominators of r-Egyptian fraction expansion for sqrt(1/2), where r(k) = 1/Prime(k).

Original entry on oeis.org

1, 2, 5, 325, 164073, 23835564403, 509747166181000498873, 590605960011761211516665913403247265840072, 493340534610970903685535778248091335992630045997033895220604001625216391426083646793

Offset: 1

Clark Kimberling, Mar 30 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) + 1/(3*2) + 1/(5*5) + 1/(7*325) + ...

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, A000040.

r[k_] := 1/Prime[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[1/2]; Table[n[x, k], {k, 1, z}]

cp's OEIS Frontend

A269999 Denominators of r-Egyptian fraction expansion for Pi - 3, where r = (1,1/2,1/3,1/4,...)

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A270001 Denominators of r-Egyptian fraction expansion for 1/e, where r = (1,1/2,1/3,1/4,...)

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A270002 Denominators of r-Egyptian fraction expansion for e - 2, where r = (1,1/2,1/3,1/4,...)

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A270314 Denominators of r-Egyptian fraction expansion for log(2), where r = (1,1/2,1/3,1/4,...).

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A270315 Denominators of r-Egyptian fraction expansion for the Euler-Mascheroni constant (Gamma-Euler constant), where r = (1,1/2,1/3,1/4,...)

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A270316 Denominators of r-Egyptian fraction expansion for (1/2)^(1/3), where r = (1,1/2,1/3,1/4,...)

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A270358 Denominators of r-Egyptian fraction expansion for (1/2)^(1/3), where r = (1, 1/2, 1/4, 1/8, ...).

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A270394 Denominators of r-Egyptian fraction expansion for sqrt(1/2), where r(k) = 1/Fibonacci(k+1).

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