A194789 -id:A194789 - cp's OEIS frontend

A194556 Decimal expansion of (9/4)^(27/8) = (27/8)^(9/4).

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

Offset: 2

Jonathan Sondow, Aug 30 2011

Positive real numbers x < y with x^y = y^x are parameterized by (x,y) = ((1 + 1/t)^t,(1 + 1/t)^(t+1)) for t > 0. For example, t = 2 gives (x,y) = (9/4,27/8). See Sondow and Marques 2010, pp. 155-157.

(9/4)^(27/8) = (27/8)^(9/4) corresponds to (4/9)^(4/9) = (8/27)^(8/27) (see A194789) under the equivalence x^y = y^x <==> (1/x)^(1/x) = (1/y)^(1/y).

			15.438887358552583183604460013074909718871494279680272412854330453294418363...

J. Sondow and D. Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae, 37 (2010), 151-164.
Index entries for algebraic numbers, degree 4

Cf. A073226 (e^e), A194557 (sqrt(3)^sqrt(27) = sqrt(27)^sqrt(3)), A194789 ((4/9)^(4/9) = (8/27)^(8/27)).

RealDigits[ (9/4)^(27/8), 10, 100] // First

-((9*ProductLog(-1, -(4/9)*log(9/4)))/(4*log(9/4))), where ProductLog is the Lambert W function, simplifies to 27/8. - Jean-François Alcover, Jun 01 2015

A258503 Decimal expansion of (64/27)^(256/81) = (256/81)^(64/27).

Original entry on oeis.org

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

Offset: 2

Jean-François Alcover, Jun 01 2015

			15.2969313436178682300313080466454951313357722002517312514576871...

Jonathan Sondow, Diego Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae 37 (2010) 151-164

Cf. A194556, A194789, A258504.

RealDigits[(64/27)^(256/81), 10, 103] // First

-((x*ProductLog(-1, -(log(x)/x)))/log(x)), replacing x with 64/27, gives 256/81 (ProductLog is the Lambert W function).

A258504 Decimal expansion of (27/64)^(27/64) = (81/256)^(81/256).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 01 2015

			0.6948233607279167955009391708983145471565914206815539940260405544...

Jonathan Sondow, Diego Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae 37 (2010) 151-164

Cf. A194556, A194789, A258503.

RealDigits[(27/64)^(27/64), 10, 105] // First

x^x = y^y <==> (1/x)^(1/y) = (1/y)^(1/x), hence, from A258503:

(64/27)^(256/81) = (256/81)^(64/27) <==> (27/64)^(27/64) = (81/256)^(81/256).

cp's OEIS Frontend

A194556 Decimal expansion of (9/4)^(27/8) = (27/8)^(9/4).

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A258503 Decimal expansion of (64/27)^(256/81) = (256/81)^(64/27).

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A258504 Decimal expansion of (27/64)^(27/64) = (81/256)^(81/256).

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