A068985 -id:A068985 - cp's OEIS frontend

A092616 Decimal expansion of e^(-1/4).

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

Offset: 0

Mohammad K. Azarian, Apr 22 2004

			0.778800783071404

Cf. A001113, A019774, A068985.

RealDigits[Surd[E,-4],10,120][[1]] (* Harvey P. Dale, Jul 07 2013 *)
Limit[Product[(k/n)^(k/n^2), {k, 1, n}], n->Infinity] (* Vaclav Kotesovec, Oct 06 2023 *)

Equals limit_{n->oo} Product_{k=1..n} (k/n)^(k/n^2). - Vaclav Kotesovec, Oct 06 2023

A092727 Decimal expansion of e^(-1/6).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Apr 22 2004

			0.84648172489061407404491739979875457688829162442705...

Cf. A001113, A019774, A068985, A092515 (reciprocal).

RealDigits[E^-(1/6),10,120][[1]] (* Harvey P. Dale, Jun 18 2012 *)

Equals lim_{x->0} (sin(x)/x)^(1/x^2). - Amiram Eldar, Jul 04 2022

A107586 Powers of e^(1/e) rounded up.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 10, 14, 19, 28, 40, 58, 83, 120, 173, 250, 360, 521, 752, 1086, 1569, 2266, 3273, 4728, 6831, 9868, 14256, 20594, 29752, 42981, 62093, 89704, 129592, 187217, 270466, 390734, 564480, 815486, 1178107, 1701973, 2458785, 3552127, 5131644, 7413521

Offset: 0

Henry Bottomley, May 16 2005

nonn

Smallest integer such that a(n)^x-x^n is nonnegative for all nonnegative reals x.

Cf. A001113, A017984, A061481, A068985, A073229.

Table[Ceiling[(E^(1/E))^n],{n,0,43}] (* James C. McMahon, Feb 12 2024 *)

A135003 Decimal expansion of 3/e.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Nov 15 2007

			1.103638323514...

G. C. Greubel, Table of n, a(n) for n = 1..2000

Cf. A001113, A068985, A165457.

RealDigits[3/E , 10, 50][[1]] (* G. C. Greubel, Sep 16 2016 *)

PARI
```
3/exp(1) \\ Michel Marcus, Sep 16 2016
```

Equals Sum_{n>=0} 1/A165457(n). - Jaume Oliver Lafont, Oct 03 2009
Equals 3/A001113 and 3*A068985. - Michel Marcus, Sep 16 2016
Equals 1 - Integral_{x=-1/e..0} W(x) dx, where W is Lambert's function. - Amiram Eldar, Jul 18 2021

Minor edits by Omar E. Pol, Oct 07 2009

A174549 a(n) = (2n-1)! + (2n)!.

Original entry on oeis.org

3, 30, 840, 45360, 3991680, 518918400, 93405312000, 22230464256000, 6758061133824000, 2554547108585472000, 1175091669949317120000, 646300418472124416000000, 418802671169936621568000000, 315777214062132212662272000000, 274094621805930760590852096000000

Offset: 1

Paul Curtz, Mar 22 2010

x*cos(x) - sin(x) = Sum_{n>=1} (-1)^n/a(n) * x^(2*n+1). - James R. Buddenhagen, Nov 21 2013

Also the number of adjacency matrices for the n-helm graph. - Eric W. Weisstein, May 25 2017

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Eric Weisstein's World of Mathematics, Adjacency Matrix.
Eric Weisstein's World of Mathematics, Helm Graph.

Cf. A001048, A000255, A068985, A143624, A165457.

[Factorial(2*n-1) + Factorial(2*n): n in [1..15]]; // Vincenzo Librandi, Aug 04 2011

A174549 := proc(n) (1+2*n)*(2*n-1)! ; end proc: # R. J. Mathar, Jan 13 2011

Table[(2*n - 1)! + (2*n)!, {n, 15}] (* T. D. Noe, Nov 21 2013 *)

a(n)=(2*n-1)!*(2*n+1) \\ Charles R Greathouse IV, Nov 21 2013

a(n) = A001048(2n) = (1+2n)*(2n-1)! = 3*A165457(n-1).
Sum_{n>=1} 1/a(n) = A068985 = 1/e = lim_{n->infinity} A000255(n-1)/A001048(n).
zeta(2*n+1) = Integral_{u=0..Pi/2} (sin(u)*log(sin(u))^(2*n+1)/(cos(u)^3))*(-2)^(2*n+1)/(n*a(n)) du. Verified for n=1 to 4 on Wolfram Alpha. - Jean-Claude Babois, Oct 28 2014
Sum_{n>=1} (-1)^(n+1)/a(n) = sin(1)-cos(1) = (-1)*A143624. - Amiram Eldar, Apr 12 2021

A270752 (r,1)-greedy sequence, where r(k) = 1/(k*e).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 262, 167395, 42355398928, 2986137074379747535250, 16334453331070842795541380956715272941358931, 334377619479874433401339085661668551899899040409749812309411639875183486098285324762070

Offset: 1

Clark Kimberling, Apr 09 2016

Let x > 0, and let r = (r(k)) be a sequence of positive irrational numbers. Let a(1) be the least positive integer m such that r(1)/m < x, and inductively let a(n) be the least positive integer m such that r(1)/a(1) + ... + r(n-1)/a(n-1) + r(n)/m < x. The sequence (a(n)) is the (r,x)-greedy sequence. We are interested in choices of r and x for which the series r(1)/a(1) + ... + r(n)/a(n) + ... converges to x. See A270744 for a guide to related sequences.

			a(1) = ceiling(r(1)) = ceiling(1/e) = ceiling(0.367...) = 1;
a(2) = ceiling(r(2)/(1 - r(1)/1)) = 1;
a(3) = ceiling(r(3)/(1 - r(1)/1 - r(2)/1)) = 1.
The first 6 terms of the series r(1)/a(1) + ... + r(n)/a(n) + ... are 0.367..., 0.551..., 0.674..., 0.766..., 0.839..., 0.901...

Cf. A001620, A270744, A068985.

$MaxExtraPrecision = Infinity; z = 16;
r[k_] := N[1/(k*E), 1000]; f[x_, 0] = x;
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; Table[n[x, k], {k, 1, z}]
N[Sum[r[k]/n[x, k], {k, 1, 18}], 200]

a(n) = ceiling(r(n)/s(n)), where s(n) = 1 - r(1)/a(1) - r(2)/a(2) - ... - r(n-1)/a(n-1).

r(1)/a(1) + ... + r(n)/a(n) + ... = 1.

A325905 Decimal expansion of 2/e^2.

Original entry on oeis.org

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

Offset: 0

José María Grau Ribas, Sep 08 2019

Decimal expansion of the asymptotic (n -> inf) probability of success in the secretary problem when the number of applicants is uniformly distributed on {1, 2, ..., n}. It means that less is known than in the basic secretary problem, so this constant is less than A068985 (and also A246665).

			0.2706705664732253837879989899...

E. L. Presman and I. M. Sonin, The best choice problem for a random number of objects, Teor. Veroyatnost. i Primenen., 17:4 (1972), 695-706; English version, Theory Probab. Appl., 17:4 (1973), 657-668.
Eric Weisstein's MathWorld, Sultan's Dowry Problem
Wikipedia, Secretary problem

Equals twice A092553.

Cf. A068985, A246665.

Mathematica
```
N[2/E^2, 100] // RealDigits // First
```

A383215 Primes p preceded and followed by gaps whose difference (absolute value) is greater than log(p).

Original entry on oeis.org

7, 29, 31, 113, 127, 139, 149, 181, 191, 199, 223, 241, 283, 307, 317, 331, 347, 419, 421, 431, 467, 521, 523, 541, 619, 641, 661, 673, 773, 809, 811, 821, 829, 853, 863, 877, 887, 907, 953, 967, 1009, 1021, 1031, 1049, 1051, 1061, 1069, 1087, 1129, 1151, 1153, 1213, 1259, 1277

Offset: 1

Alain Rocchelli, Apr 19 2025

nonn

Primes prime(k) such that abs(prime(k-1)-2*prime(k)+prime(k+1)) > log(prime(k)), where log is the natural logarithm.

a(n) ~ prime(round(n*e)) as n tends to infinity, where e is Euler's number.

			7 is a term because abs(5-2*7+11)=2 and log(7)=1.9459.
29 is a term because abs(23-2*29+31)=4 and log(29)=3.3673.

Cf. A036263, A383216, A068985.

Select[Prime[Range[2,206]],Abs[NextPrime[#,-1]-2#+NextPrime[#]]>Log[#]&] (* James C. McMahon, Apr 29 2025 *)

forprime(P=3, 1300, my(M=P-precprime(P-1), Q=nextprime(P+1)-P, AR1=min(M,Q), AR2=max(M,Q), AR0=log(P)); if(AR2-AR1>AR0, print1(P,", ")));

Limit_{n->oo} n / PrimePi(a(n)) = 1/e (A068985).

A092618 Decimal expansion of e^(-1/5).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Apr 22 2004

			0.81873075307798

Cf. A001113, A019774, A068985.

RealDigits[Power[E, (-5)^-1],10,120][[1]] (* Harvey P. Dale, Jan 27 2012 *)

A092750 Decimal expansion of e^(-1/7).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Apr 22 2004

			0.866877899750...

Cf. A001113, A019774, A068985.

evalf(1/exp(1/7), 124);  # Alois P. Heinz, Apr 04 2020

RealDigits[N[Exp[-1/7], 112]][[1]] (* Georg Fischer, Apr 04 2020 *)

a(104) corrected by Georg Fischer, Apr 04 2020

cp's OEIS Frontend

A092616 Decimal expansion of e^(-1/4).

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A092727 Decimal expansion of e^(-1/6).

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A107586 Powers of e^(1/e) rounded up.

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A135003 Decimal expansion of 3/e.

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A174549 a(n) = (2*n-1)! + (2*n)!.

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Magma

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A270752 (r,1)-greedy sequence, where r(k) = 1/(k*e).

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Formula

A325905 Decimal expansion of 2/e^2.

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Mathematica

A383215 Primes p preceded and followed by gaps whose difference (absolute value) is greater than log(p).

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A174549 a(n) = (2n-1)! + (2n)!.