A269993 -id:A269993 - cp's OEIS frontend

A270487 Denominators of r-Egyptian fraction expansion for (1/2)^(1/3), where r(k) = 1/Prime(k).

1, 2, 2, 6, 29, 860, 626907, 1582796431872, 4577382865450526674426008, 77218331531088831524423800072197013265311322482652, 10410509369911993512345323774444196964795747018426948027297775848734862056109801420845614477793011811

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) + 1/(3*2) + 1/(5*2) + 1/(7*6) + ...

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

r(k) = 1/prime(k);
f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
a(k, x=(1/2)^(1/3)) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 31 2016

A270517 Denominators of r-Egyptian fraction expansion for sqrt(1/2), where r(k) = 1/k!.

Original entry on oeis.org

2, 3, 5, 6, 52, 668, 171510, 4590170768, 17103459833953822083, 31906466290986600582512428032058109695, 237271596693541800921324673318278335675822001026279366213434934428597656224

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/(1*2) + 1/(2*3) + 1/(6*5) + 1/(24*6) + ...

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, A000142, A010503.

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 = Sqrt[1/2]; Table[n[x, k], {k, 1, z}]

r(k) = 1/k!;
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 31 2016

A270527 Denominators of r-Egyptian fraction expansion for (1/2)^(1/3), where r(k) = 1/k!.

Original entry on oeis.org

2, 2, 4, 21, 168, 10754, 25461498, 105205312405537, 2273436544813042470905435068, 580632014636885174037652548241171956049642213022500047, 105076738483143967759563061000636154401568577693011463452250666394865203888381724797435152416096091560375615

Offset: 1

Clark Kimberling, Apr 02 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/(1*2) + 1/(2*2) + 1/(6*4) + 1/(24*21) + ...

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, A000142.

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}]

A270546 Denominators of r-Egyptian fraction expansion for sqrt(1/2), where r(k) = 1/(2k-1).

Original entry on oeis.org

2, 2, 5, 325, 200533, 65627675599, 22975481891957121466348, 1958997403653886589078102754522745217186637162, 141280756113351994103874857935521871912536028357392961997286697261498102983722388787617517574

Offset: 1

Clark Kimberling, Apr 02 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/(1*2) + 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, A005408, A010503.

r[k_] := 1/(2k-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/(2*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, Apr 03 2016

A270557 Denominators of r-Egyptian fraction expansion for (1/2)^(1/3), where r(k) = 1/(2k-1).

Original entry on oeis.org

2, 2, 2, 6, 35, 1828, 87102089, 9369260399911997, 79759690931475868535017424372273, 6278545782421133501164266118042557416295332543123744442037840298

Offset: 1

Clark Kimberling, Apr 03 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/(1*2) + 1/(3*2) + 1/(5*2) + 1/(7*6) + ...

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, A005408.

r[k_] := 1/(2k-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}]

A270580 Denominators of r-Egyptian fraction expansion for sqrt(1/2), where r(k) = 1/(k+1).

Original entry on oeis.org

1, 2, 7, 43, 2233, 5100361, 40162526999265, 25631935256046376027999327548, 973579151885397220180400699680033378225854987721289580493, 20355636044566797478491707686529410726939762602606154042023303177125252037523393842033572704449460687246942494130101

Offset: 1

Clark Kimberling, Apr 03 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/(4*7) + 1/(5*43) + ...

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+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}]

A270582 Denominators of r-Egyptian fraction expansion for sqrt(2) - 1, where r(k) = 1/(k+1).

Original entry on oeis.org

2, 3, 5, 65, 6529, 136091233, 41625259047416909, 2189507051227161558033650829868135, 75931290362065676573711484986356332365619562746656079489987281066955

Offset: 1

Clark Kimberling, Apr 03 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/(2*2) + 1/(3*3) + 1/(4*5) + 1/(5*65) + ...

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+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[2] - 1; Table[n[x, k], {k, 1, z}]

A270584 Denominators of r-Egyptian fraction expansion for golden ratio - 1, where r(k) = 1/(k+1).

Original entry on oeis.org

1, 3, 37, 1204, 21029921, 425355555167420, 439183524292095499600664584581, 240317442633783387248198509182959563857071128274317237128901, 1816763565571992723556609635427913847146292698536599340539742991592182627925499061514094793847919952134648005118828414904

Offset: 1

Clark Kimberling, Apr 03 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) + 1/(3*3) + 1/(4*37) + 1/(5*1204) + ...

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+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 = GoldenRatio - 1; Table[n[x, k], {k, 1, z}]

A270591 Denominators of r-Egyptian fraction expansion for (1/2)^(1/3), where r(k) = 1/(k+1).

Original entry on oeis.org

1, 2, 2, 99, 12204, 249462465, 93524017020207705, 8528549813750403709101762452858246, 70071914165301390868341700110703069865385640933927590404095892463912

Offset: 1

Clark Kimberling, Apr 04 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) + 1/(3*2) + 1/(4*2) + 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.

r[k_] := 1/(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}]

A270348 Denominators of r-Egyptian fraction expansion for sqrt(1/3), where r = (1,1/2,1/4,1/8,...)

Original entry on oeis.org

2, 7, 43, 1161, 796510, 1101781866330, 648667164391834988511313, 521313118065995695198529265268104396429334449023, 177042477384698216444912803612486097958997328262217304760270340328784709181787835108737458616981

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(1/3) = 1/2 + 1/(2*7) + 1/(4*43) + ...

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 = Sqrt[1/3]; 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=sqrt(1/3)) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 18 2016

cp's OEIS Frontend

A270487 Denominators of r-Egyptian fraction expansion for (1/2)^(1/3), where r(k) = 1/Prime(k).

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A270517 Denominators of r-Egyptian fraction expansion for sqrt(1/2), where r(k) = 1/k!.

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A270527 Denominators of r-Egyptian fraction expansion for (1/2)^(1/3), where r(k) = 1/k!.

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A270546 Denominators of r-Egyptian fraction expansion for sqrt(1/2), where r(k) = 1/(2k-1).

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A270557 Denominators of r-Egyptian fraction expansion for (1/2)^(1/3), where r(k) = 1/(2k-1).

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A270580 Denominators of r-Egyptian fraction expansion for sqrt(1/2), where r(k) = 1/(k+1).

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A270582 Denominators of r-Egyptian fraction expansion for sqrt(2) - 1, where r(k) = 1/(k+1).

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A270584 Denominators of r-Egyptian fraction expansion for golden ratio - 1, where r(k) = 1/(k+1).

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