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A269993 Denominators of r-Egyptian fraction expansion for sqrt(1/2), where r = (1,1/2,1/3,1/4,...) 99
2, 3, 9, 74, 8098, 101114070, 10080916639334518, 234737156891222571756748160861129, 104728182461244680288139397973895577148266725366426255244889745185 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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 of x shown above (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

LINKS

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

EXAMPLE

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

MATHEMATICA

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}]

PROG

(PARI) 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

CROSSREFS

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

Sequence in context: A120032 A126884 A054544 * A132537 A251543 A248236

Adjacent sequences:  A269990 A269991 A269992 * A269994 A269995 A269996

KEYWORD

nonn,frac,easy

AUTHOR

Clark Kimberling, Mar 15 2016

STATUS

approved

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Last modified December 8 01:42 EST 2019. Contains 329850 sequences. (Running on oeis4.)