cp's OEIS Frontend

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

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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

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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