A068985 -id:A068985 - cp's OEIS frontend

A092560 Decimal expansion of e^(-5).

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

Offset: 0

Mohammad K. Azarian, Apr 09 2004

			0.006737946999085467096636048423148424248849585027...

Cf. A019774, A001113, A068985.

First[RealDigits[E^-5, 10, 100, -1]] (* Paolo Xausa, Feb 09 2025 *)

A092577 Decimal expansion of e^(-6).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Apr 09 2004

			0.0024787521766663

Cf. A019774, A001113, A068985.

Join[{0,0},RealDigits[1/E^6,10,120][[1]]] (* Harvey P. Dale, Sep 29 2013 *)

A092578 Decimal expansion of e^(-7).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Apr 09 2004

			0.00091188196555451620800313608440928262647372452743605...

Cf. A019774, A001113, A068985.

First[RealDigits[E^-7, 10, 100, -1]] (* Paolo Xausa, Feb 09 2025 *)

A125313 Decimal expansion of 2*exp(-gamma).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Dec 08 2006

			1.12291896713377033964828642958176157353142077385030633630831815209...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.3, Landau-Ramanujan constant, p. 100.

G. C. Greubel, Table of n, a(n) for n = 1..10000
A. Granville, Harald Cramér and the distribution of prime numbers, Scandinavian Actuarial Journal 1: 12-28, (1995) DOI:10.1080/03461238.1995.10413946.
Bernard Montaron, Exponential prime sequences, arXiv:2011.14653 [math.NT], 2020.
Simon Plouffe, Plouffe's Inverter.
S. K. Wilson and B. R. Duffy, An asymptotic analysis of small holes in thin fluid layers, Journal of Engineering Mathematics, July 1996, Volume 30, Issue 4, pp 445-457.

R:= RealField(100); 2*Exp(-EulerGamma(R)); // G. C. Greubel, Sep 05 2018

RealDigits[2*Exp[-EulerGamma], 10, 111][[1]]

default(realprecision, 100); 2*exp(-Euler) \\ G. C. Greubel, Sep 05 2018

Equals 2*A080130, 2*A001113^(-A001620) and 2/A073004 = 2/A068985^A001620.
Equals A088540 * A088541. - Jean-François Alcover, Jun 04 2014
Equals exp(A002162 - A001620). - John W. Nicholson, Apr 03 2015

A181589 Least value of n such that P(n) - 1/e < 10^(-i), i=1,2,3... . P(n) = (n/(n+1))^(n-1) the probability of a random forest on n be a tree.

Original entry on oeis.org

6, 56, 553, 5519, 55183, 551820, 5518192, 55181917, 551819162, 5518191618, 55181916176, 551819161758, 5518191617572, 55181916175717, 551819161757164, 5518191617571636, 55181916175716349, 551819161757163483

Offset: 1

Washington Bomfim, Oct 31 2010

nonn

The probability P(n) = A000169(n)/A000272(n+1). It is known that lim_{n->inf}p(n) = 1/e. (See Flajolet and Sedgewick link, pp 632, where we can find a function of the number of components k).
Both P(n) and the probability that a permutation on n objects be a derangement tend to 1/e when n rises to infinity. So the events a random forest be a tree and a random permutation be a derangement become equiprobable as n tends to infinity.
The probability P(n) approaches 1/e quite slowly as this sequence shows. See image clicking the first link.

			a(1) = 6, a(2) = 56, so for n in the interval 6...55 if we use 1/e as the probability P, we make an error less than 10^(-1). In general if n is in the interval a(i), ... , a(i+1)-1, this error is less than 10^(-i).

Flajolet and Sedgewick, Analytic Combinatorics
Wikipedia, Graph of probabilities
Wikipedia, Derangements

Cf. A000169, A000272, A068985, A000166, A181590.

A185362 Decimal expansion of 2^(1/e).

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 26 2012

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A068985, decimal expansion of 1/e; A003417 with an initial 0 added, continued fraction of 1/e.

evalf(2^exp(-1), 100); # Wesley Ivan Hurt, Jun 28 2017

Mathematica
```
RealDigits[N[2^(1/E),200]][[1]]
```

2^exp(-1) \\ Charles R Greathouse IV, Apr 25 2016

A195326 Numerators of fractions leading to e - 1/e (A174548).

Original entry on oeis.org

0, 2, 2, 7, 7, 47, 47, 5923, 5923, 426457, 426457, 15636757, 15636757, 7318002277, 7318002277, 1536780478171, 1536780478171, 603180793741, 603180793741, 142957467201379447, 142957467201379447

Offset: 0

Paul Curtz, Oct 12 2011

The sequence of approximations of exp(1) obtained by truncating the Taylor series of exp(x) after n terms is A061354(n)/A061355(n) = 1, 2, 5/2, 8/3, 65/24, ...
A Taylor series of exp(-1) is 1, 0, 1/2, 1/3, 3/8, ... and (apart from the first 2 terms) given by A000255(n)/A001048(n). Subtracting both sequences term by term we obtain a series for exp(1) - exp(-1) = 0, 2, 2, 7/3, 7/3, 47/20, 47/20, 5923/2520, 5923/2520, 426457/181440, 426457/181440, ... which defines the numerators here.
Each second of the denominators (that is 3, 2520, 19958400, ...) is found in A085990 (where each third term, that is 60, 19958400, ...) is to be omitted.
This numerator sequence here is basically obtained by doubling entries of A051397, A009628, A087208, or A186763, caused by the standard associations between cosh(x), sinh(x) and exp(x).

			a(0) =  1  -  1;
a(1) =  2  -  0;
a(2) = 5/2 - 1/2.

Cf. A001113, A068985, A197222, A197223.

taylExp1 := proc(n)
        add(1/j!,j=0..n) ;
end proc:
A000255 := proc(n)
        if n <=1 then
                1;
        else
                n*procname(n-1)+(n-1)*procname(n-2) ;
        end if;
end proc:
A001048 := proc(n)
        n!+(n-1)! ;
end proc:
A195326 := proc(n)
        if n = 0 then
                0;
        elif n =1 then
                2;
        else
                taylExp1(n) -A000255(n-2)/A001048(n-1);
        end if;
          numer(%);
end proc:
seq(A195326(n),n=0..20) ; # R. J. Mathar, Oct 14 2011

Material meant to be placed in other sequences removed by R. J. Mathar, Oct 14 2011

A270401 Denominators of r-Egyptian fraction expansion for 1/e, where r(k) = 1/Fibonacci(k+1).

Original entry on oeis.org

3, 15, 275, 306142, 119655359789, 11580087075793732204662, 149024368678486978900547818363959440890696944, 23508494642625759146052819702452314132546312046986774534693830017181700550956750715996250

Offset: 1

Clark Kimberling, Mar 22 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.

See A269993 for a guide to related sequences.

			1/e = 1/3 + 1/(2*15) + 1/(3*275) + ...

Clark Kimberling, Table of n, a(n) for n = 1..11
Eric Weisstein's World of Mathematics, Egyptian Fraction
Index entries for sequences related to Egyptian fractions

Cf. A269993, A000045, A068985.

r[k_] := 1/Fibonacci[k+1]; 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 = 1/E; Table[n[x, k], {k, 1, z}]

r(k) = 1/fibonacci(k+1);
f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
a(k, x=exp(-1)) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 22 2016

A270524 Denominators of r-Egyptian fraction expansion for 1/e, where r(k) = 1/k!.

Original entry on oeis.org

3, 15, 138, 8259, 239711437, 11635520651781427, 27093570249697895705902826783521, 172213812234178083415955164135638766722297909315047111470339241

Offset: 1

Clark Kimberling, Mar 30 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.

See A269993 for a guide to related sequences.

			1/e = 1/(1*3) + 1/(2*15) + 1/(6*138) + 1/(24*8259) + ...

Clark Kimberling, Table of n, a(n) for n = 1..11
Eric Weisstein's World of Mathematics, Egyptian Fraction
Index entries for sequences related to Egyptian fractions

Cf. A269993, A000142, A068985.

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 = 1/E; Table[n[x, k], {k, 1, z}]

r(k) = 1/k!;
f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
a(k, x=exp(-1)) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 31 2016

A270553 Denominators of r-Egyptian fraction expansion for 1/e, where r(k) = 1/(2k-1).

Original entry on oeis.org

3, 10, 165, 218673, 75510967206, 14666670996451472494064, 318033435047744040119174255756277946082958110, 222562499295932133989982996162129528076446080094832884826693648678455802606574139206041317

Offset: 1

Clark Kimberling, Apr 02 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.

See A269993 for a guide to related sequences.

			1/e = 1/(1*3) + 1/(3*10) + 1/(5*165) + 1/(7*218673) + ...

Clark Kimberling, Table of n, a(n) for n = 1..11
Eric Weisstein's World of Mathematics, Egyptian Fraction
Index entries for sequences related to Egyptian fractions

Cf. A269993, A005408, A068985.

r[k_] := 1/(2k-1); 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 = 1/E; Table[n[x, k], {k, 1, z}]

r(k) = 1/(2*k-1);
f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
a(k, x=exp(-1)) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Apr 03 2016

cp's OEIS Frontend

A092560 Decimal expansion of e^(-5).

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A092577 Decimal expansion of e^(-6).

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A092578 Decimal expansion of e^(-7).

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A125313 Decimal expansion of 2*exp(-gamma).

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A181589 Least value of n such that P(n) - 1/e < 10^(-i), i=1,2,3... . P(n) = (n/(n+1))^(n-1) the probability of a random forest on n be a tree.

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A185362 Decimal expansion of 2^(1/e).

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Maple

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A195326 Numerators of fractions leading to e - 1/e (A174548).

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Maple

Extensions

A270401 Denominators of r-Egyptian fraction expansion for 1/e, where r(k) = 1/Fibonacci(k+1).

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A270524 Denominators of r-Egyptian fraction expansion for 1/e, where r(k) = 1/k!.

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