A006525 -id:A006525 - cp's OEIS frontend

A269993 Denominators of r-Egyptian fraction expansion for sqrt(1/2), where r = (1,1/2,1/3,1/4,...)

2, 3, 9, 74, 8098, 101114070, 10080916639334518, 234737156891222571756748160861129, 104728182461244680288139397973895577148266725366426255244889745185

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.
Guide to related sequences:
r(k)               x               denominators
1               sqrt(1/2)             A069139
1               sqrt(1/3)             A144983
1               sqrt(2) - 1           A006487
1               sqrt(3) - 1           A118325
1               tau - 1               A117116
1               1/Pi                  A006524
1               Pi-3                  A001466
1               1/e                   A006526
1                e - 2                A006525
1               log(2)                A118324
1               Euler constant        A110820
1               (1/2)^(1/3)           A269573
.
1/k             sqrt(1/2)             A269993
1/k             sqrt(1/3)             A269994
1/k             sqrt(2) - 1           A269995
1/k             sqrt(3) - 1           A269996
1/k             tau - 1               A269997
1/k             1/Pi                  A269998
1/k             Pi-3                  A269999
1/k             1/e                   A270001
1/k             e - 2                 A270002
1/k             log(2)                A270314
1/k             Euler constant        A270315
1/k             (1/2)^(1/3)           A270316
.
Using the 12 choices for x shown above (that is, sqrt(1/2) to (1/2)^(1/3)), the denominator sequence of the r-Egyptian fraction for x appears for each of the following sequences (r(k)):
r(k) = 1 (see above)
r(k) = 1/k (see above)
r(k) = 2^(1-k):  A270347-A270358
r(k) = 1/Fibonacci(k+1):  A270394-A270405
r(k) = 1/prime(k):  A270476-A270487
r(k) = 1/k!:  A270517-A270527 (A000027 for x = e - 2)
r(k) = 1/(2k-1):  A270546-A270557
r(k) = 1/(k+1):  A270580-A270591

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

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. A269573, A069139, A270347, A270394, A270476, A270517, A270546, A270580.

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;
x = sqrt(1/2);
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, 10, print1(n(x, k),", ")) \\ Indranil Ghosh, Mar 27 2017, translated from Mathematica code

A001466 Denominators of greedy Egyptian fraction expansion of Pi - 3.

Original entry on oeis.org

8, 61, 5020, 128541455, 162924332716605980, 28783052231699298507846309644849796, 871295615653899563300996782209332544845605756266650946342214549769447

Offset: 1

N. J. A. Sloane

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

			Pi - 3 = 1/8 + 1/61 + 1/5020 + 1/128541455 + ... .

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Simon Plouffe, Table of n, a(n) for n = 1..10 (There is a limit of about 1000 digits on the size of numbers in b-files)
Mohammad K. Azarian, An Expression for Pi, Problem #870, College Mathematics Journal, Vol. 39, No. 1, January 2008, p. 66. Solution appeared in Vol. 40, No. 1, January 2009, pp. 62-64.
K. R. R. Gandhi, Edifice of the real numbers by alternating series, International Journal of Mathematical Archive-3(9), 2012, 3277-3280. - From _N. J. A. Sloane_, Jan 02 2013
Simon Plouffe, Table of n, a(n) for n = 1..14
H. P. Robinson, Letter to N. J. A. Sloane, Sep 1975
H. E. Salzer, The approximation of numbers as sums of reciprocals, Amer. Math. Monthly, 54 (1947), 135-142.
Eric Weisstein's World of Mathematics, Egyptian Fraction
Wikipedia, Greedy algorithm for Egyptian fractions
J. W. Wrench, Jr., Letter to N. J. A. Sloane, Mar 27 1974
Index entries for sequences related to the number Pi

See A182257, A224230 for other versions of this sequence.

Cf. A006525 (similar for e-2).

lst={};k=N[(Pi-3),1000];Do[s=Ceiling[1/k];AppendTo[lst,s];k=k-1/s,{n,12}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 02 2009 *)

x = Pi - 3;
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

A144835 Denominators of an Egyptian fraction for 1/zeta(2) = 0.607927101854... (A059956).

Original entry on oeis.org

2, 10, 127, 18838, 522338493, 727608914652776081, 990935377560451600699026552443764271, 1223212384013602554473872691328685513734082755736750146553750539914774364

Offset: 1

Artur Jasinski, Sep 22 2008

			1/zeta(2) = 0.607927101854... = 1/2 + 1/10 + 1/127 + 1/18838 + ...

Amiram Eldar, Table of n, a(n) for n = 1..11
Mohammad K. Azarian, Sylvester's Sequence and the Infinite Egyptian Fraction Decomposition of 1, Problem 958, College Mathematics Journal, Vol. 42, No. 4, September 2011, p. 330. Solution published in Vol. 43, No. 4, September 2012, pp. 340-342.
Eric Weisstein's World of Mathematics, Egyptian Fraction.
Index entries for sequences related to Egyptian fractions.

Cf. A001466, A006487, A006524, A006525, A006526, A059956, A069139, A110820, A117116, A118323, A118324, A118325.

a = {}; k = N[1/Zeta[2], 1000]; Do[s = Ceiling[1/k]; AppendTo[a, s]; k = k - 1/s, {n, 1, 10}]; a

x=1/zeta(2); while(x, t=1\x+1; print1(t", "); x -= 1/t) \\ Charles R Greathouse IV, Nov 08 2013

A144984 Denominators of an Egyptian fraction for 1/sqrt(5) (A020762).

Original entry on oeis.org

3, 9, 362, 148807, 432181530536, 615828580117398011389583, 385329014801969222669766835659574445455872858297

Offset: 1

Artur Jasinski, Sep 28 2008

Jinyuan Wang, Table of n, a(n) for n = 1..11

Cf. A020762, A069139, A006487, A006526, A006525, A006524, A001466, A110820, A117116, A118323, A118324, A118325, A144835, A132480-A132574, A069261, A144984-A145003.

a = {}; k = N[1/Sqrt[5], 1000]; Do[s = Ceiling[1/k]; AppendTo[a, s]; k = k - 1/s, {n, 1, 10}]; a

A145003 Denominators of an Egyptian fraction for 1/sqrt(29) = 0.185695338... (A020786).

Original entry on oeis.org

6, 53, 6221, 891830563, 950677235679298964, 2245647960428048728674383451656707058, 11636905679093503238901947768600244923435901955366623291532461461126244496

Offset: 1

Artur Jasinski, Sep 28 2008

Cf. A069139, A006487, A006526, A006525, A006524, A001466, A110820, A117116, A118323, A118324, A118325, A144835, A132480-A132574, A069261, A144984-A145003.

a = {}; k = N[1/Sqrt[29], 1000]; Do[s = Ceiling[1/k]; AppendTo[a, s]; k = k - 1/s, {n, 1, 10}]; a

A073422 e = 1/a(0)+1/a(1)+1/a(2)+1/a(3)+... with each term a(n) being a positive or negative integer chosen so as to minimize the absolute difference between e and the partial sum.

Original entry on oeis.org

1, 1, 2, 5, 55, 9999, 3620211522, -26596490130011501642, 6462025287494698350477135653962718736877, -532695733923048954868620962844302990205269539900643893905567041090276924142488084

Offset: 0

Henry Bottomley, Jul 30 2002

sign

			a(4)=55 since e-(1/1)-(1/1)-(1/2)-(1/5)=0.0182818... is closer to 1/55=0.0181818... than to 1/54=0.0185185...

Cf. A001467, A006525, A014015.

A144983 Denominators of greedy Egyptian fraction for 1/sqrt(3) (A020760).

Original entry on oeis.org

2, 13, 2341, 41001128, 3352885935529869, 17147396444547741051849884001699, 1847333322606272250132077006229901193256553492442739965269739579

Offset: 1

Artur Jasinski, Sep 28 2008

Amiram Eldar, 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. A001466, A006487, A006524, A006525, A006526, A020760, A069139, A069261, A110820, A117116, A118323, A118324, A118325, A144835, A132480-A132574, A144984-A145003.

a = {}; k = N[1/Sqrt[3], 1000]; Do[s = Ceiling[1/k]; AppendTo[a, s]; k = k - 1/s, {n, 1, 10}]; a

A142725 Denominators of an Egyptian fraction for 1/Sqrt[17] = 0.242535625...

Original entry on oeis.org

5, 24, 1151, 6727710, 97954001297811, 12083213443785578998604325741, 2111557350230332542969297514824119073134312726162508784857, 5126406954746155312559668571658555244727150562238830979161154018392336359308299948544053564102183773577991816755308

Offset: 1

Artur Jasinski, Sep 28 2008

A069139, A006487, A006526, A006525, A006524, A001466, A110820, A117116, A118323, A118324, A118325, A144835, A132480-A132574, A069261, A144984-A145003

a = {}; k = N[1/Sqrt[17], 1000]; Do[s = Ceiling[1/k]; AppendTo[a, s]; k = k - 1/s, {n, 1, 10}]; a (*Artur Jasinski*)

A142726 Denominators of an Egyptian fraction for 1/Sqrt[20] = 0.2236067977...

Original entry on oeis.org

5, 43, 2850, 9380555, 131539825706327, 25568462906010064277774504354, 1702783284378767791750994476557209698496292570221862357616, 9282809298390896944529722953873240985108041182275536393531898614770319137100914187360035180181565645720539192811580

Offset: 1

Artur Jasinski, Sep 28 2008

A069139, A006487, A006526, A006525, A006524, A001466, A110820, A117116, A118323, A118324, A118325, A144835, A132480-A132574, A069261, A144984-A145003

a = {}; k = N[1/Sqrt[20], 1000]; Do[s = Ceiling[1/k]; AppendTo[a, s]; k = k - 1/s, {n, 1, 10}]; a (*Artur Jasinski*)

A144985 Denominators of an Egyptian fraction for 1/Sqrt[6]=0.408248290463863...

Original entry on oeis.org

3, 14, 287, 484228, 624850913463, 832896370765715143490072, 7620764031777359266114991754446899201236457828088, 74466937067918173179787895367258766085493130434332689333832927329763999409894621431449951498850730

Offset: 1

Artur Jasinski, Sep 28 2008

A069139, A006487, A006526, A006525, A006524, A001466, A110820, A117116, A118323, A118324, A118325, A144835, A132480-A132574, A069261, A144984-A145003

a = {}; k = N[1/Sqrt[6], 1000]; Do[s = Ceiling[1/k]; AppendTo[a, s]; k = k - 1/s, {n, 1, 10}]; a (*Artur Jasinski*)

cp's OEIS Frontend

A269993 Denominators of r-Egyptian fraction expansion for sqrt(1/2), where r = (1,1/2,1/3,1/4,...)

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A001466 Denominators of greedy Egyptian fraction expansion of Pi - 3.

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A144835 Denominators of an Egyptian fraction for 1/zeta(2) = 0.607927101854... (A059956).

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A144984 Denominators of an Egyptian fraction for 1/sqrt(5) (A020762).

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A145003 Denominators of an Egyptian fraction for 1/sqrt(29) = 0.185695338... (A020786).

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A073422 e = 1/a(0)+1/a(1)+1/a(2)+1/a(3)+... with each term a(n) being a positive or negative integer chosen so as to minimize the absolute difference between e and the partial sum.

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A144983 Denominators of greedy Egyptian fraction for 1/sqrt(3) (A020760).

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Mathematica

A142725 Denominators of an Egyptian fraction for 1/Sqrt[17] = 0.242535625...

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Mathematica

A142726 Denominators of an Egyptian fraction for 1/Sqrt[20] = 0.2236067977...

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Mathematica

A144985 Denominators of an Egyptian fraction for 1/Sqrt[6]=0.408248290463863...

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