A269993 -id:A269993 - cp's OEIS frontend

A006525 Denominators of greedy Egyptian fraction for e - 2.

2, 5, 55, 9999, 3620211523, 25838201785967533906, 3408847366605453091140558218322023440765

Offset: 1

A greedy Egyptian fraction is also called a Sylvester expansion. - Robert FERREOL, May 02 2020

			e - 2 = 1/2 + 1/5 + 1/55 + 1/9999 + ... . - _Jon E. Schoenfield_, Dec 26 2014

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..11 [a(11) corrected by _Georg Fischer_, Jun 22 2020]
H. P. Robinson, Letter to N. J. A. Sloane, Sep 1975
Eric Weisstein's World of Mathematics, Egyptian Fraction
Wikipedia, Greedy algorithm for Egyptian fractions
Index entries for sequences related to Egyptian fractions

Cf. A006526, A269993.

Cf. A001466 (similar for Pi-3).

lst={};k=N[E-2,1000000];Do[s=Ceiling[1/k];AppendTo[lst,s];k=k-1/s,{n,12}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 02 2009 *)

x = exp(1) - 2;
f(x, k) = if(k<1, x, f(x, k - 1) - 1/n(x, k));
n(x, k) = ceil(1/f(x, k - 1));
for(k = 1, 7, print1(n(x, k), ", ")) \\ Indranil Ghosh, Mar 27 2017

a(n) = ceiling(1/(e - 2 - Sum_{j=0..n-1} 1/a(j))). - Jon E. Schoenfield, Dec 26 2014

More terms from Herman P. Robinson

Offset changed to 1 by Indranil Ghosh, Mar 27 2017

A270744 (r,1)-greedy sequence, where r(k) = 1/tau^k, where tau = golden ratio.

Original entry on oeis.org

1, 2, 2, 3, 4, 32, 1065, 2038968, 5977146319204, 36314862033946243071181679, 1028280647188781709727717632740627249617427013751977, 958046899855070460620234639622630375078362220775180051610386376308132568342498992099474472596860400289

Offset: 1

Clark Kimberling, Apr 07 2016

Let x > 0, and let r = (r(k)) be a sequence of positive irrational numbers.  Let a(1) be the least positive integer m such that r(1)/m < x, and inductively let a(n) be the least positive integer m such that r(1)/a(1) + ... + r(n-1)/a(n-1) + r(n)/m < x.  The sequence (a(n)) is the (r,x)-greedy sequence.  We are interested in choices of r and x for which the series r(1)/a(1) + ... + r(n)/a(n) + ... converges to x.  (The same algorithm is used to generate sequences listed at A269993.)
Guide to related sequences:
  x     r(k)
 1/tau^k             A270744
 k/tau^k             A270745
 2/e^k               A270746
 4/Pi^k              A270747
 2/log(k+1)          A270748
 k/log(k+1)          A270749
 1/(k*log(k+1))      A270750
 1/(k*tau)           A270751
 1/(k*e)             A270752
 1/(k*sqrt(2))       A270916

			a(1) = ceiling(r(1)) = ceiling(1/tau) = ceiling(0.618...) = 1;
a(2) = ceiling(r(2)/(1 - r(1)/1)) = 2;
a(3) = ceiling(r(3)/(1 - r(1)/1 - r(2)/2)) = 2.
The first 6 terms of the series r(1)/a(1) + ... + r(n)/a(n) + ... are 0.618..., 0.809..., 0.927..., 0.975..., 0.998..., 0.999967... .

Cf. A001620, A270745, A094214, A269993.

$MaxExtraPrecision = Infinity; z = 13;
r[k_] := N[1/GoldenRatio^k, 1000]; f[x_, 0] = x;
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; Table[n[x, k], {k, 1, z}]
N[Sum[r[k]/n[x, k], {k, 1, 13}], 200]

a(n) = ceiling(r(n)/s(n)), where s(n) = 1 - r(1)/a(1) - r(2)/a(2) - ... - r(n-1)/a(n-1).

r(1)/a(1) + ... + r(n)/a(n) + ... = 1.

A270714 Decimal expansion of (1/2)^(1/3).

Original entry on oeis.org

7, 9, 3, 7, 0, 0, 5, 2, 5, 9, 8, 4, 0, 9, 9, 7, 3, 7, 3, 7, 5, 8, 5, 2, 8, 1, 9, 6, 3, 6, 1, 5, 4, 1, 3, 0, 1, 9, 5, 7, 4, 6, 6, 6, 3, 9, 4, 9, 9, 2, 6, 5, 0, 4, 9, 0, 4, 1, 4, 2, 8, 8, 0, 9, 1, 2, 6, 0, 8, 2, 5, 2, 8, 1, 2, 1, 0, 9, 5, 8, 6, 6, 3, 6, 7, 7

Offset: 0

Clark Kimberling, Mar 22 2016

Let c = (1/2)^(1/3). A sphere of radius c*r has half the volume of a sphere of radius r. - Rick L. Shepherd, Aug 12 2016

Let c = (1/2)^(1/3). The relative maximum of xy(x+y)=1 is (c,-1/c^2). - Clark Kimberling, Oct 05 2020

			0.79370052598409973737585281963615413019574666394992650490414288091260825...

Clark Kimberling, Table of n, a(n) for n = 0..10000

Cf. A002580, A005480, A235362, A269993.

Mathematica
```
RealDigits[(1/2)^(1/3), 10, 200][[1]]
```
PARI
```
(1/2)^(1/3) \\ Altug Alkan, Mar 22 2016
```

Equals 1/A002580 = A002580*A235362 = A005480*A020761. [corrected and expanded by Rick L. Shepherd, Aug 12 2016]

Equals Product_{k>=1} (1 + (-1)^k/(3*k+1)). - Amiram Eldar, Aug 10 2020

A270347 Denominators of r-Egyptian fraction expansion for sqrt(1/2), where r = (1,1/2,1/4,1/8,...)

Original entry on oeis.org

2, 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(1/2) = 1/2 + 1/(2*3) + 1/(4*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_] := 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[1/2]; 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=sqrt(1/2)) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 18 2016

A269573 Denominators of r-Egyptian fraction expansion for (1/2)^(1/3), where r = (1,1,1,1,1,...)

Original entry on oeis.org

2, 4, 23, 4500, 23314202, 703143261541584, 580028504455491926110281336263, 471554575224119231041268294704259548817134505334232514876247

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.

			(1/2)^(1/3) = 1/2 + 1/4 + 1/23 + ...

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 (guide to related sequences).

r[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 = 2^(-1/3); Table[n[x, k], {k, 1, z}]  (* A269573 *)

A269994 Denominators of r-Egyptian fraction expansion for sqrt(1/3), where r = (1,1/2,1/3,1/4,...)

Original entry on oeis.org

2, 7, 57, 3391, 10010183, 588972486242552, 961457184347597076119863109462, 2244227167765735741796211572067153905745156229769919746729015

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.

			sqrt(1/3) = 1/2 + 1/(2*7) + 1/(3*57) + ...

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 = Sqrt[1/3]; Table[n[x, k], {k, 1, z}]

r(k) = 1/k;
x = sqrt(1/3);
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 27 2017, translated from Mathematica code

A269995 Denominators of r-Egyptian fraction expansion for sqrt(2) - 1, where r = (1,1/2,1/3,1/4,...)

Original entry on oeis.org

3, 7, 36, 1300, 2206054, 14887222782418, 292542996759533035472424790, 7282957087563143077864043818232331102110274520711753058, 259880230781524461939787525796521055875618560291171401151227648777033604862236784108033156713828890456025177451

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.

			sqrt(2) - 1 = 1/(2*3) + 1/(3*7) + 1/(4*36) + ...

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

A269996 Denominators of r-Egyptian fraction expansion for sqrt(3) - 1, where r = (1,1/2,1/3,1/4,...)

Original entry on oeis.org

2, 3, 6, 26, 939, 800567, 626897816036, 732632470241183632257841, 31706715561023122142248280773186018287458544854469, 1666726692230759969765850044548001173784581299264219742879080654883940143766478552206863259848365362

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.

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

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[3] - 1; Table[n[x, k], {k, 1, z}]

A269997 Denominators of r-Egyptian fraction expansion for -1 + golden ratio, where r = (1,1/2,1/3,1/4,...)

Original entry on oeis.org

2, 5, 19, 511, 224138, 60658204540, 203857858414658884506671, 65699957103246706854223474912465037343245580906, 3942313430901049708832516976840058495554562175116278047675351101544028510870033057494673090034

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.

			tau - 1 = 1/2 + 1/(2*5) + 1/(3*19) + ...

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

A269998 Denominators of r-Egyptian fraction expansion for 1/Pi, where r = (1,1/2,1/3,1/4,...)

Original entry on oeis.org

4, 8, 58, 3984, 22875462, 931267108879599, 1031674577884217945682977326053, 1260295551033259417770370489346530643885445465368122822066849

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/Pi = 1/4 + 1/(2*8) + 1/(3*58) + ...

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/Pi; Table[n[x, k], {k, 1, z}]

r(k) = 1/k;
x = 1/Pi;
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

cp's OEIS Frontend

A006525 Denominators of greedy Egyptian fraction for e - 2.

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A270744 (r,1)-greedy sequence, where r(k) = 1/tau^k, where tau = golden ratio.

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A270714 Decimal expansion of (1/2)^(1/3).

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A270347 Denominators of r-Egyptian fraction expansion for sqrt(1/2), where r = (1,1/2,1/4,1/8,...)

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A269573 Denominators of r-Egyptian fraction expansion for (1/2)^(1/3), where r = (1,1,1,1,1,...)

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A269994 Denominators of r-Egyptian fraction expansion for sqrt(1/3), where r = (1,1/2,1/3,1/4,...)

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A269995 Denominators of r-Egyptian fraction expansion for sqrt(2) - 1, where r = (1,1/2,1/3,1/4,...)

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A269996 Denominators of r-Egyptian fraction expansion for sqrt(3) - 1, where r = (1,1/2,1/3,1/4,...)