A309652 -id:A309652 - cp's OEIS frontend

A073229 Decimal expansion of e^(1/e).

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

Offset: 1

Rick L. Shepherd, Jul 22 2002

e^(1/e) = 1/((1/e)^(1/e)) (reciprocal of A072364).
Let w(n+1)=A^w(n); then w(n) converges if and only if (1/e)^e <= A <= e^(1/e) (see the comments in A073230) for initial value w(1)=A. If A=e^(1/e) then lim_{n->infinity} w(n) = e. - Benoit Cloitre, Aug 06 2002; corrected by Robert FERREOL, Jun 12 2015
x^(1/x) is maximum for x = e and the maximum value is e^(1/e). This gives an interesting and direct proof that 2 < e < 4 as 2^(1/2) < e^(1/e) > 4^(1/4) while 2^(1/2) = 4^(1/4). - Amarnath Murthy, Nov 26 2002
For large n, A234604(n)/A234604(n-1) converges to e^(1/e). - Richard R. Forberg, Dec 28 2013
Value of the unique base b > 0 for which the exponential curve y=b^x and its inverse y=log_b(x) kiss each other; the kissing point is (e,e). - Stanislav Sykora, May 25 2015
Actually, there is another base with such property, b=(1/e)^e with kiss point (1/e,1/e). - Yuval Paz, Dec 29 2018
The problem of finding the maximum of f(x) = x^(1/x) was posed and solved by the Swiss mathematician Jakob Steiner (1796-1863) in 1850. - Amiram Eldar, Jun 17 2021

			1.44466786100976613365833910859...

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 35.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Ovidiu Furdui, Problem 11982, The American Mathematical Monthly, Vol. 124, No. 5 (2017), p. 465; A Limit of a Power of a Sum, Solution to Problem 11982 by Roberto Tauraso, ibid., Vol. 126, No. 2 (2019), p. 187.
Simon Plouffe, exp(1/e).
Jonathan Sondow and Diego Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae, Vol. 37 (2010), pp. 151-164; see Definition 4.1 on p. 158.
Jacob Steiner, Über das größte Product der Theile oder Summanden jeder Zahl, Crelle, Vol. 40 (1850), pp. 208; alternative link.
Eric Weisstein's World of Mathematics, Steiner's Problem.

Cf. A001113 (e), A068985 (1/e), A073230 ((1/e)^e), A072364 ((1/e)^(1/e)), A073226 (e^e).
Cf. A093157, A103476.
Cf. A038051, A086331, A252782, A270593, A270917, A270923, A277473, A309652, A332408.

evalf[110](exp(exp(-1))); # Muniru A Asiru, Dec 29 2018

Mathematica
```
RealDigits[ E^(1/E), 10, 110] [[1]]
```
PARI
```
exp(1)^exp(-1)
```

Equals 1 + Integral_{x = 1/e..1} (1 + log(x))/x^x dx = 1 - Integral_{x = 0..1/e} (1 + log(x))/x^x dx. - Peter Bala, Oct 30 2019
Equals Sum_{k>=0} exp(-k)/k!. - Amiram Eldar, Aug 13 2020
Equals lim_{x->oo} (Sum_{n>=1} (x/n)^n)^(1/x) (Furdui, 2017). - Amiram Eldar, Mar 26 2022

A326986 G.f.: B(x)B(x^2)B(x^3)*..., where B(x) is g.f. of A000312.

Original entry on oeis.org

1, 1, 5, 29, 266, 3163, 46994, 827107, 16828741, 388308078, 10017853262, 285720195351, 8926575094978, 303172417424680, 11121259586618456, 438207141286916539, 18458204444260001120, 827690809585441201775, 39365349178064541861252, 1979267564496263599093676

Offset: 0

Vaclav Kotesovec, Aug 10 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..386

Cf. A000312, A096161, A309652, A326985.

B:= proc(n) option remember; n^n end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=1,
      B(n), add(b(j, 1)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 23 2019

nmax = 20; CoefficientList[Series[Product[1+Sum[k^k*x^(j*k), {k, 1, nmax/j}], {j, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ n^n.

cp's OEIS Frontend

A073229 Decimal expansion of e^(1/e).

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A326986 G.f.: B(x)B(x^2)B(x^3)*..., where B(x) is g.f. of A000312.

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cp's OEIS Frontend

A073229 Decimal expansion of e^(1/e).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A326986 G.f.: B(x)*B(x^2)*B(x^3)*..., where B(x) is g.f. of A000312.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A326986 G.f.: B(x)B(x^2)B(x^3)*..., where B(x) is g.f. of A000312.