A019739 -id:A019739 - cp's OEIS frontend

A244009 Decimal expansion of 1 - log(2).

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

Offset: 0

Franklin T. Adams-Watters, Jun 17 2014

Fraction of numbers which are sqrt-smooth, see A048098 and A063539. - Charles R Greathouse IV, Jul 14 2014

Asymptotic survival probability in the 100 prisoners problem. - Alois P. Heinz, Jul 08 2022

			0.30685281944005469058276787854...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, pp. 43-44.

Peter Bala, A continued fraction expansion for the constant 1 - log(2)
Donald E. Knuth and Luis Trabb Pardo, Analysis of a simple factorization algorithm, Theoretical Computer Science 3:3 (1976), pp. 321-348.
The Ramanujan Machine, Using algorithms to discover new mathematics.
Wikipedia, 100 prisoners problem
Index entries for transcendental numbers

Cf. A002162, A019739, A024168, A100381, A135002, A188859.

Essentially the same digits as A239354.

f:= sum(1/(2*k*(2*k+1)), k=1..infinity):
s:= convert(evalf(f, 140), string):
seq(parse(s[i+1]), i=1..106);  # Alois P. Heinz, Jun 17 2014

RealDigits[1-Log[2],10,120][[1]] (* Harvey P. Dale, Sep 23 2016 *)

1-log(2) \\ Charles R Greathouse IV, Jul 14 2014

Equals Sum_{k>=0} 1/(2*k*(2*k+1)) = A239354 + 1/4 = A188859/2.
From Amiram Eldar, Aug 07 2020: (Start)
Equals Sum_{k>=1} 1/(k*(k+1)*2^k) = Sum_{k>=2} 1/A100381(k).
Equals Sum_{k>=2} (-1)^k * zeta(k)/2^k.
Equals Integral_{x=1..oo} 1/(x^2 + x^3) dx. (End)
Equals log(e/2) = log(A019739) = -log(2/e) = -log(A135002). - Wolfdieter Lang, Mar 04 2022
Equals lim_{n->oo} A024168(n)/n!. - Alois P. Heinz, Jul 08 2022
Equals 1/(4 - 4/(7 - 12/(10 - ... - 2*n*(n-1)/((3*n+1) - ...)))) (an equivalent continued fraction for 1 - log(2) was conjectured by the Ramanujan machine). - Peter Bala, Mar 04 2024
Equals Sum_{k>=1} zeta(2*k)/((2*k + 1)*2^(2*k-1)) (see Finch). - Stefano Spezia, Nov 02 2024

A279607 Beatty sequence for e/2; i.e., a(n) = floor(n*e/2).

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 9, 10, 12, 13, 14, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 35, 36, 38, 39, 40, 42, 43, 44, 46, 47, 48, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 72, 73, 74, 76, 77, 78, 80, 81, 82, 84, 85, 86, 88, 89

Offset: 1

Clark Kimberling, Dec 16 2016

The complement is A279608, the Beatty sequence for e/(e - 2).

Clark Kimberling, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Beatty sequences

Cf. A019739 (e/2), A279608.

r = E/2; s = r/(r - 1); z = 10000;
Table[Floor[n*r], {n, 1, z}] ;(* A279607 *)
Table[Floor[n*s], {n, 1, z}] ;(* A279608 *)

e = exp(1);
for(n=1, 100, print1(floor(n*e/2),", ")) \\ Indranil Ghosh, Mar 30 2017

import math
from mpmath import mp
mp.dps=100
print([int(math.floor(n*e/2)) for n in range(1, 101)]) # Indranil Ghosh, Mar 30 2017

A006083 Continued fraction for e/2.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 1, 3, 3, 3, 1, 3, 1, 3, 5, 3, 1, 5, 1, 3, 7, 3, 1, 7, 1, 3, 9, 3, 1, 9, 1, 3, 11, 3, 1, 11, 1, 3, 13, 3, 1, 13, 1, 3, 15, 3, 1, 15, 1, 3, 17, 3, 1, 17, 1, 3, 19, 3, 1, 19, 1, 3, 21, 3, 1, 21, 1, 3, 23, 3, 1, 23, 1, 3, 25, 3, 1, 25, 1, 3, 27, 3, 1, 27, 1, 3, 29, 3, 1, 29, 1, 3, 31, 3

Offset: 1

N. J. A. Sloane, Jeffrey Shallit

			1.359140914229522617680143735... = 1 + 1/(2 + 1/(1 + 1/(3 + 1/(1 + ...)))). - _Harry J. Smith_, May 10 2009

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 2, p. 601.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Harry J. Smith, Table of n, a(n) for n = 1..20000
G. Xiao, Contfrac
Index entries for continued fractions for constants
Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,0,1,0,0,-1).

Cf. A019739 = Decimal expansion. - Harry J. Smith, May 10 2009

ContinuedFraction[E/2, 94] (* Jean-François Alcover, Apr 01 2011 *)
Join[{1, 2},LinearRecurrence[{0, 0, 1, 0, 0, 1, 0, 0, -1},{1, 3, 1, 1, 1, 3, 3, 3, 1},92]] (* Ray Chandler, Sep 03 2015 *)

{ allocatemem(932245000); default(realprecision, 55000); x=contfrac(exp(1)/2); for (n=1, 20000, write("b006083.txt", n, " ", x[n])); } \\ Harry J. Smith, May 10 2009

Vec(x*(1+2*x+x^2+2*x^3-x^4-3*x^6+x^8-x^10) / ((1-x)^2*(1+x)*(1-x+x^2)*(1+x+x^2)^2) + O(x^50)) \\ Colin Barker, May 16 2016

a(1)=1, a(2)=2, a(3)=1, a(4)=3, a(5)=1, a(6)=1, a(7)=1, a(8)=3, then for k>=1 a(6*k+3)=a(6*k+6)=2*k+1, a(6*k+4)=a(6*k+8)=3, a(6*k+5)=a(6*k+7)=1. - Benoit Cloitre, Apr 08 2003
From Colin Barker, May 16 2016: (Start)
a(n) = a(n-3)+a(n-6)-a(n-9) for n>9.
G.f.: x*(1+2*x+x^2+2*x^3-x^4-3*x^6+x^8-x^10) / ((1-x)^2*(1+x)*(1-x+x^2)*(1+x+x^2)^2).
(End)

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 20 2003

A257775 Decimal expansion of (e/2)^2.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, May 12 2015

The coefficient a of the unique parabola y = a*x^2 which, at some x > 0, kisses the exponential function y = exp(x). The kissing point coordinates are (2,e^2).

			1.847264024732662556807606865143751953295078892637961831021781955630...

Stanislav Sykora, Table of n, a(n) for n = 1..2000

Cf. A001113, A019739, A257776, A257777.

RealDigits[Exp[2]/4, 10, 120][[1]] (* Amiram Eldar, Jun 26 2023 *)

PARI
```
exp(2)/4
```

A058298 Triangle n!/(n-k), 1 <= k < n, read by rows.

Original entry on oeis.org

2, 3, 6, 8, 12, 24, 30, 40, 60, 120, 144, 180, 240, 360, 720, 840, 1008, 1260, 1680, 2520, 5040, 5760, 6720, 8064, 10080, 13440, 20160, 40320, 45360, 51840, 60480, 72576, 90720, 120960, 181440, 362880, 403200, 453600, 518400, 604800, 725760, 907200, 1209600, 1814400, 3628800

Offset: 2

Leroy Quet, Dec 07 2000

Together with 1, numbers n such that n divides k! if and only if k! >= n. - Charles R Greathouse IV, Aug 16 2016

			Triangle begins:
      2;
      3,     6;
      8,    12,    24;
     30,    40,    60,   120;
    144,   180,   240,   360,   720;
    840,  1008,  1260,  1680,  2520,   5040;
   5760,  6720,  8064, 10080, 13440,  20160,  40320;
  45360, 51840, 60480, 72576, 90720, 120960, 181440, 362880;
  ...

Andrew Howroyd, Table of n, a(n) for n = 2..1276
Florian Unger and Jonathan Krebs, MCMC Sampling of Directed Flag Complexes with Fixed Undirected Graphs, arXiv:2309.02938 [math.ST], 2023.

Columns k=1..5 are A001048(n-1), A052747, A052759, A052778, A052794.
Row sums are A052881.
Cf. A019739. A077012.

Flatten[Table[n!/(n-k),{n,2,10},{k,n-1}]] (* Harvey P. Dale, Jul 23 2014 *)

PARI
```
T(n,k)={if(kAndrew Howroyd, Aug 08 2020
```

Sum_{n>=2} Sum_{k=1..n-1} 1/T(n, k) = e/2 (A019739). - Amiram Eldar, Jun 29 2025

A334397 Decimal expansion of (e - 2)/e.

Original entry on oeis.org

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

Offset: 0

Daniel Hoyt, Apr 26 2020

			0.2642411176571153568089524596770782651...

M. L. Glasser, A note on Beukers's and related integrals, Amer. Math. Monthly 126(4) (2019), 361-363.

Cf. A001008, A001113, A002805, A019739, A135002, A309419, A334396.

RealDigits[1 - 2/E, 10, 100][[1]] (* Alonso del Arte, Apr 26 2020 *)

1 - 2/exp(1) \\ Michel Marcus, May 01 2020

Equals Integral_{x=0..1} x/e^x dx.
Equals 1 - A135002.
Equals 1/A309419.
Equals -Integral_{x=0..1, y=0..1} x*y/(exp(x*y)*log(x*y)) dx dy. (Apply Theorem 1 or Theorem 2 of Glasser (2019) to the above integral.) - Petros Hadjicostas, Jun 30 2020
From Amiram Eldar, Aug 05 2020: (Start)
Equals Sum_{k>=0} (-1)^k/(k! * (k+2)).
Equals Sum_{k>=1} 1/((2*k)! * (k+1)).
Equals Sum_{k>=1} (-1)^k * k^2 * H(k)/k!, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. (End)

A084148 Numerators of terms in the Pippenger product.

Original entry on oeis.org

2, 8, 1152, 1605632, 43913893117952, 98583626709555431615548620800, 197241992148713072661201501950348880945923403897735704916000768

Offset: 1

Eric W. Weisstein, May 15 2003

G. C. Greubel, Table of n, a(n) for n = 1..10
Nicholas Pippenger, An infinite product for e, The American Mathematical Monthly, Vol. 87, No. 5 (1980), p. 391.
Eric Weisstein's World of Mathematics, Pippenger Product.

Cf. A019739, A084149 (denominators).

F:=Factorial;
A084148:= func< n | n eq 1 select 2 else Round(Numerator( 2^(2^n -1)*(F(2^(n-1)))^6 / ((F(2^n))^2 * (F(2^(n-2)))^4) )) >;
[A084148(n): n in [1..10]]; // G. C. Greubel, Oct 13 2022

a[n_] := Numerator[((2^(n - 1) - 1)!!*(2^n)!!/((2^(n - 1))!!*(2^n - 1)!!))^2/2]; Array[a, 7] (* Amiram Eldar, Apr 10 2022 *)

f=factorial
def A084148(n): return 2 if (n==1) else numerator( 2^(2^n -1)*(f(2^(n-1)))^6 / ((f(2^n))^2 * (f(2^(n-2)))^4) )
[A084148(n) for n in range(1,10)] # G. C. Greubel, Oct 13 2022

From Amiram Eldar, Apr 10 2022: (Start)
a(n) =  numerator(((2^(n-1)-1)!!*(2^n)!!/((2^(n-1))!!*(2^n-1)!!))^2/2).
Product_{n>=1} (a(n)/A084149(n))^(1/2^n) = e/2 (A019739). (End)
a(n) = numerator( 2^(2^n -1)*((2^(n-1))!)^6 / (((2^n)!)^2 * ((2^(n-2))!)^4) ), with a(1) = 2. - G. C. Greubel, Oct 13 2022

A196533 Decimal expansion of 15*e.

Original entry on oeis.org

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

Offset: 2

R. J. Mathar, Oct 03 2011

			40.7742274268856785304043120702... = 15*A001113 = 30*A019739.

L. B. W. Jolley, Summation of Series, Dover, 1961, eq (101) on page 20.

SetDefaultRealField(RealField(100)); 15*Exp(1); // Vincenzo Librandi, Apr 05 2020

Maple
```
evalf(15*exp(1));
```

RealDigits[15 E,10,120][[1]] (* Harvey P. Dale, Aug 24 2019 *)

15*exp(1) \\ Charles R Greathouse IV, Apr 21 2016

Equals sum(j>=1, j^3/(j-1)! ).

Equals 10 * sum(n>=1, A000217(n)/n! ). - Richard R. Forberg, Jul 14 2013

A022766 Ordered sequence of distinct terms of the form floor(x^i * floor(x^j)), i,j >= 0, where x = e/2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 20, 21, 23, 25, 27, 28, 29, 31, 34, 37, 38, 39, 43, 46, 47, 50, 51, 52, 53, 54, 58, 63, 64, 68, 69, 71, 72, 73, 79, 86, 87, 93, 94, 97, 98, 99, 108, 116, 119, 126, 128, 129, 132, 133, 134, 135, 146, 158

Offset: 1

Clark Kimberling

nonn

Cf. A019739 (e/2).

Offset corrected by Sean A. Irvine, May 20 2019

A378942 Decimal expansion of (1/sqrt(Pi) + e*erfc(-1))/2.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Dec 11 2024

			2.7865848321550198766289520248877912002691926567823...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 4.6, p. 262.

Cf. A001113, A019739, A087197, A087198, A099286, A222392.

RealDigits[(1/Sqrt[Pi]+E Erfc[-1])/2,10,100][[1]]

Equals (1 + e*sqrt(Pi)*(1 + erf(1)))/(2*sqrt(Pi)).

Equals A222392 / 2. - Amiram Eldar, Feb 15 2025

cp's OEIS Frontend

A244009 Decimal expansion of 1 - log(2).

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A279607 Beatty sequence for e/2; i.e., a(n) = floor(n*e/2).

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A006083 Continued fraction for e/2.

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A257775 Decimal expansion of (e/2)^2.

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A058298 Triangle n!/(n-k), 1 <= k < n, read by rows.

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A334397 Decimal expansion of (e - 2)/e.

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A084148 Numerators of terms in the Pippenger product.

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