A007676 -id:A007676 - cp's OEIS frontend

A003417 Continued fraction for e.

2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, 14, 1, 1, 16, 1, 1, 18, 1, 1, 20, 1, 1, 22, 1, 1, 24, 1, 1, 26, 1, 1, 28, 1, 1, 30, 1, 1, 32, 1, 1, 34, 1, 1, 36, 1, 1, 38, 1, 1, 40, 1, 1, 42, 1, 1, 44, 1, 1, 46, 1, 1, 48, 1, 1, 50, 1, 1, 52, 1, 1, 54, 1, 1, 56, 1, 1, 58, 1, 1, 60, 1, 1, 62, 1, 1, 64, 1, 1, 66

Offset: 0

N. J. A. Sloane

This is also the Engel expansion for 3*exp(1/2)/2 - 1/2. - Gerald McGarvey, Aug 07 2004
Sorted with duplicate terms dropped, this is A004277, 1 together with the positive even numbers. - Alonso del Arte, Jan 27 2012
From Peter Bala, Nov 26 2019: (Start)
Related continued fractions expansions:
2*e = [5; 2, 3, 2, 3, 1, 2, 1, 3, 4, 3, 1, 4, 1, 3, 6, 3, 1, 6, ..., 1, 3, 2*n, 3, 1, 2*n, ...].
(1/2)*e = [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, 2*n + 1, 3, 1, 2*n + 1, ...].
4*e = [10, 1, 6, 1, 7, 2, 7, 2, 7, 1, 1, 1, 7, 3, 7, 1, 2, 1, 7, 4, 7, 1, 3, 1, 7, 5, 7, 1, 4, ..., 1, 7, n+1, 7, 1, n, ...].
(1/4)*e = [0, 1, 2, 8, 3, 1, 1, 1, 1, 7, 1, 1, 2, 1, 1, 1, 2, 7, 1, 2, 2, 1, 1, 1, 3, 7, 1, 3, 2, 1, 1, 1, 4, 7, 1, 4, 2, ..., 1, 1, 1, n, 7, 1, n, 2, ...]. (End)

			2.718281828459... = 2 + 1/(1 + 1/(2 + 1/(1 + 1/(1 + ...))))

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 186.
CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 88.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3.2.
Jay R. Goldman, The Queen of Mathematics, 1998, p. 70.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 0..9999
Khalil Ayadi, Chiheb Ben Bechir, and Maher Saadaoui, Continued Fractions with Predictable Patterns and Transcendental Numbers, J. Int. Seq. (2025) Vol. 28, Art. No. 25.1.4.
Thomas Baruchel and Carsten Elsner, On error sums formed by rational approximations with split denominators, arXiv preprint arXiv:1602.06445 [math.NT], 2016.
Henry Cohn, A short proof of the simple continued fraction expansion of e, Amer. Math. Monthly, 113 (No. 1, 2006), 57-62. [JSTOR] and arXiv:math/0601660 [math.NT], 2006.
Stephen Crowley, Integral Transforms of the Harmonic Sawtooth Map, The Riemann Zeta Function, and Fractal Strings, vixra:1202.0079 v2, 2012. [Wayback Machine link]
Francesco Dolce and Pierre-Adrien Tahay, Column representation of Sturmian words in cellular automata, Czech Technical University (Prague, Czechia, 2022).
W. R. Harmon, Letter to N. J. A. Sloane, Sep 1974
MathOverflow, What is the effect of adding 1/2 to a continued fraction?, 2013.
Keith Matthews, Finding the continued fraction of e^(l/m)
Sophie Morier-Genoud and Valentin Ovsienko, On q-deformed real numbers, arXiv:1908.04365 [math.QA], 2019.
C. D. Olds, The simple continued fraction expansion of e, Am. Math. Monthly 77 (9) (1970) 968-974.
Thomas J. Osler, A proof of the continued fraction expansion of e^(1/M), Amer. Math. Monthly, 113 (No. 1, 2006), 62-66.
Oskar Perron, Die Lehre von den Kettenbrüchen, 2nd ed., Teubner, Leipzig, 1929, p. 134.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Ward Whitt, Weirdness in CTMC's, Notes for Course IEOR 6711: Stochastic Models I, [PDF], 2012. - From _N. J. A. Sloane_, Jan 03 2013
Eric Weisstein's World of Mathematics, e Continued Fraction.
Gang Xiao, Contfrac.
Yaohui Zhu, Kaiming Sun, Zhengdong Luo, and Lingfeng Wang, Progressive Self-Learning for Domain Adaptation on Symbolic Regression of Integer Sequences, Proc. 39th AAAI Conf. Artif. Intel. (2025) Vol. 39, No. 1, 1692-1699. See p. 1698.
Index entries for continued fractions for constants.
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Cf. A001113, A007676, A007677, A001204, A058282, A005131.
Cf. A006083, A006084, A006085, A081750.
Run lengths of A342991.

numtheory[cfrac](exp(1),100,'quotients'); # Jani Melik, May 25 2006
A003417:=(2+z+2*z**2-3*z**3-z**4+z**6)/(z-1)**2/(z**2+z+1)**2; # Simon Plouffe in his 1992 dissertation

ContinuedFraction[E, 100] (* Stefan Steinerberger, Apr 07 2006 *)
a[n_] := KroneckerDelta[1, n] + 2 n/3 - (2 n - 3)/3 DirichletCharacter[3, 1, n]; Table[a[n], {n, 1, 20}] (* Enrique Pérez Herrero, Feb 23 2013 *)
Table[Piecewise[{{2, n == 0}, {2 (n + 1)/3, Mod[n, 3] == 2}}, 1], {n, 0, 120}] (* Eric W. Weisstein, Jan 05 2019 *)
Join[{2}, LinearRecurrence[{0, 0, 2, 0, 0, -1}, {1, 2, 1, 1, 4, 1}, 120]] (* Eric W. Weisstein, Jan 05 2019 *)
Join[{2}, Table[(2 (n + 4) + (1 - 2 n) Cos[2 n Pi/3] + Sqrt[3] (1 - 2 n) Sin[2 n Pi/3])/9, {n, 120}]] (* Eric W. Weisstein, Jan 05 2019 *)
Join[{2}, Flatten[Table[{1, 2n, 1}, {n, 40}]]] (* Harvey P. Dale, Jan 21 2020 *)

contfrac(exp(1)) \\ Alexander R. Povolotsky, Feb 23 2008

{ allocatemem(932245000); default(realprecision, 25000); x=contfrac(exp(1)); for (n=1, 10000, write("b003417.txt", n-1, " ", x[n])); } \\ Harry J. Smith, Apr 14 2009

A003417(n)=if(n%3<>2,1+(n==0),(n+1)/3*2) \\ M. F. Hasler, May 01 2013

def A003417(n): return 2 if n == 0 else 1 if n % 3 != 2 else (n+1)//3<<1 # Chai Wah Wu, Jul 27 2022

def eContFracTrio(n: Int): List[Int] = List(1, 2 * n, 1)
2 +: ((1 to 40).map(eContFracTrio).flatten) // Alonso del Arte, Nov 22 2020, with thanks to Harvey P. Dale

From Paul Barry, Jun 27 2006: (Start)
G.f.: (2 + x + 2*x^2 - 3*x^3 - x^4 + x^6)/(1 - 2*x^3 + x^6).
a(n) = 0^n + Sum{k = 0..n} 2*sin(2*Pi*(k - 1)/3)*floor((2*k - 1)/3)/sqrt(3). [Corrected and simplified by Jianing Song, Jan 05 2019] (End)
a(n) = 2*a(n-3) - a(n-6), n >= 7. - Philippe Deléham, Feb 10 2009
G.f.: 1 + U(0) where U(k)= 1 + x/(1 - x*(2*k + 1)/(1 + x*(2*k + 1) - 1/((2*k + 1) + 1 - (2*k + 1)*x/(x + 1/U(k+1))))); (continued fraction, 5-step). - Sergei N. Gladkovskii, Oct 07 2012
a(3*n-1) = 2*n, a(0) = 2, a(n) = 1 otherwise (i.e., for n+1 > 1, not a multiple of 3). - M. F. Hasler, May 01 2013
E.g.f.: First derivative of (2/9)*exp(x)*(x + 3) + (2/9)*exp(-x/2)*(2*x*cos((sqrt(3)/2)*x+2*Pi/3) - 3*cos((sqrt(3)/2)*x)) + x. - Jianing Song, Jan 05 2019
a(n) = floor(1/(n+1))-(floor(n/3)-floor((n+1)/3))*(2*n-1)/3+1. - Aaron J Grech, Sep 06 2024
Sum_{n>=1} (-1)^(n+1)/a(n) = 1 - log(2)/2. - Amiram Eldar, May 03 2025

Offset changed by Andrew Howroyd, Aug 07 2024

A007677 Denominators of convergents to e.

Original entry on oeis.org

1, 1, 3, 4, 7, 32, 39, 71, 465, 536, 1001, 8544, 9545, 18089, 190435, 208524, 398959, 4996032, 5394991, 10391023, 150869313, 161260336, 312129649, 5155334720, 5467464369, 10622799089, 196677847971, 207300647060, 403978495031, 8286870547680, 8690849042711

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

Same as A113874 without its first two terms. - Jonathan Sondow, Aug 16 2006

			2, 3, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71, 1264/465, 1457/536, 2721/1001, 23225/8544, 25946/9545, 49171/18089, 517656/190435, 566827/208524, 1084483/398959, 13580623/4996032, 14665106/5394991, 28245729/10391023, 410105312/150869313, 438351041/161260336, 848456353/312129649, ...

CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 88.
W. J. LeVeque, Fundamentals of Number Theory. Addison-Wesley, Reading, MA, 1977, p. 240.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Eric W. Weisstein, Table of n, a(n) for n = 0..1000 (first 200 terms from T. D. Noe)
L. Bayon, P. Fortuny, J. M. Grau, M. M. Ruiz, M. A. Oller-Marcen, The Best-or-Worst and the Postdoc problems with random number of candidates, arXiv:1809.06390 [math.PR], 2018.
E. B. Burger, Diophantine Olympics and World Champions: Polynomials and Primes Down Under, Amer. Math. Monthly, 107 (Nov. 2000), 822-829.
C. Elsner, Series of Error Terms for Rational Approximations of Irrational Numbers, J. Int. Seq. 14 (2011) # 11.1.4.
C. Elsner, M. Stein, On Error Sum Functions Formed by Convergents of Real Numbers, J. Int. Seq. 14 (2011) # 11.8.6
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641 (article), 114 (2007) 659 (addendum).
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, arXiv:0704.1282 [math.HO], 2007-2010.
J. Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), II, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010; arXiv:0709.0671 [math.NT], 2007-2009.
Eric Weisstein's World of Mathematics, e Continued Fraction
Eric Weisstein's World of Mathematics, Sultan's Dowry Problem

Cf. A007676 (numerators of convergents to e).

Cf. A003417 (continued fraction of e).

Digits := 60: E := exp(1); convert(evalf(E),confrac,50,'cvgts'): cvgts;

Denominator[Convergents[E, 40]] (* T. D. Noe, Oct 12 2011 *)
Denominator[Table[Piecewise[{
   {Hypergeometric1F1[-1 - n/3, -1 - (2 n)/3, 1]/Hypergeometric1F1[-(n/3), -1 - (2 n)/3, -1], Mod[n, 3] == 0},
   {Hypergeometric1F1[1/3 (-2 - n), -(2/3) (2 + n), 1]/Hypergeometric1F1[1/3 (-2 - n), -(2/3) (2 + n), -1], Mod[n, 3] == 1},
   {Hypergeometric1F1[1/3 (-1 - n), 1 - (2 (4 + n))/3, 1]/Hypergeometric1F1[1/3 (-4 - n), 1 - (2 (4 + n))/3, -1], Mod[n, 3] == 2}
}], {n, 0, 30}]] (* Eric W. Weisstein, Sep 09 2013 *)
Table[Piecewise[{
    {(1 + (2 n)/3)!/(n/3)! Hypergeometric1F1[-(n/3), -1 - (2 n)/3, -1], Mod[n, 3] == 0},
    {((2 (2 + n))/3)!/((2 + n)/3)! Hypergeometric1F1[1/3 (-2 - n), -(2/3) (2 + n), -1], Mod[n, 3] == 1},
    {((4 + n) (5/3 + (2 n)/3)! )/(3 ((4 + n)/3)!) Hypergeometric1F1[1/3 (-4 - n), 1 - (2 (4 + n))/3, -1], Mod[n, 3] == 2}
}], {n, 0, 30}]  (* Eric W. Weisstein, Sep 10 2013 *)

A143382 Numerator of Sum_{k=0..n} 1/k!!.

Original entry on oeis.org

1, 2, 5, 17, 71, 121, 731, 1711, 41099, 370019, 740101, 2713789, 1206137, 423355111, 846710651, 1814380259, 203210595443, 12654139763, 531473870981, 43758015399281, 525096184837561, 441080795274037, 22054039763790029

Offset: 0

Jonathan Vos Post, Aug 11 2008

Denominators are A143383. A143382(n)/A143383(n) is to A007676(n)/A007676(n) as double factorials are to factorials. A143382/A143383 fractions begin:
n numerator/denominator
1/0!! = 1/1
1/0!! + 1/1!! = 2/1
1/0!! + 1/1!! + 1/2!! = 5/2
1/0!! + 1/1!! + 1/2!! + 1/3!! = 17/6
1/0!! + 1/1!! + 1/2!! + 1/3!! + 1/4!! = 71/24
1/0!! + 1/1!! + 1/2!! + 1/3!! + 1/4!! + 1/5!! = 121/40
1/0!! + 1/1!! + 1/2!! + 1/3!! + 1/4!! + 1/5!! + 1/6!! = 731/240
The series converges to sqrt(e) + sqrt((e*Pi)/2)*erf(1/sqrt(2)) = 3.0594074053425761445... whose decimal expansion is given by A143280. The analogs of A094007 and A094008 are determined by 2 being the only prime denominator in the convergents to the sum of reciprocals of double factorials and prime numerators beginning: a(1) = 2, a(2) = 5, a(3) = 17, a(4) = 71, a(15) = 1814380259, a(19) = 43758015399281, a(21) = 441080795274037, a(23) = 867081905243923.

			a(3) = 17 because 1/0!! + 1/1!! + 1/2!! + 1/3!! = 17/6.
a(15) = 1814380259 because 1814380259/593049600 = 1/1 + 1/1 + 1/2 + 1/3 + 1/8 + 1/15 + 1/48 + 1/105 + 1/384 + 1/945 + 1/3840 + 1/10395 + 1/46080 + 1/135135 + 1/645120 + 1/2027025.

G. C. Greubel, Table of n, a(n) for n = 0..500
Eric W. Weisstein, Double Factorial. Gives formula for limit of series, which was independently derived by Max Alekseyev.

Cf. A006882 (n!!), A094007, A143280 (m(2)), A143383 (denominators).

[n le 0 select 1 else Numerator( 1 + (&+[ 1/(0 + (&*[k-2*j: j in [0..Floor((k-1)/2)]])) : k in [1..n]]) ): n in [0..25]]; // G. C. Greubel, Mar 28 2019

Table[Numerator[Sum[1/k!!, {k, 0, n}]], {n, 0, 30}] (* G. C. Greubel, Mar 28 2019 *)
Accumulate[1/Range[0,30]!!]//Numerator (* Harvey P. Dale, May 19 2023 *)

vector(25, n, n--; numerator(sum(k=0,n, 1/prod(j=0,floor((k-1)/2), (k - 2*j)) ))) \\ G. C. Greubel, Mar 28 2019

[numerator(sum( 1/product((k - 2*j) for j in (0..floor((k-1)/2)))   for k in (0..n))) for n in (0..25)] # G. C. Greubel, Mar 28 2019

Numerators of Sum_{k=0..n} 1/k!! = Sum_{k=0..n} 1/A006882(k).

A143383 Denominator of Sum_{k=0..n} 1/k!!.

Original entry on oeis.org

1, 1, 2, 6, 24, 40, 240, 560, 13440, 120960, 241920, 887040, 394240, 138378240, 276756480, 593049600, 66421555200, 4136140800, 173717913600, 14302774886400, 171633298636800, 144171970854912, 7208598542745600, 283414985441280

Offset: 0

Jonathan Vos Post, Aug 11 2008

Numerators are A143382. A143382(n)/A143383(n) is to A007676(n)/A007676(n) as double factorials are to factorials. A143382/A143383 fractions begin:
n numerator/denominator
1/0!! = 1/1
1/0!! + 1/1!! = 2/1
1/0!! + 1/1!! + 1/2!! = 5/2
1/0!! + 1/1!! + 1/2!! + 1/3!! = 17/6
1/0!! + 1/1!! + 1/2!! + 1/3!! + 1/4!! = 71/24
1/0!! + 1/1!! + 1/2!! + 1/3!! + 1/4!! + 1/5!! = 121/40
1/0!! + 1/1!! + 1/2!! + 1/3!! + 1/4!! + 1/5!! + 1/6!! = 731/240
The series converges to sqrt(e) + sqrt((e*Pi)/2)*erf(1/sqrt(2)) = 3.0594074053425761445... whose decimal expansion is given by A143280. The analogs of A094007 and A094008 are determined by 2 being the only prime denominator in the convergents to the sum of reciprocals of double factorials and prime numerators beginning: a(1) = 2, a(2) = 5, a(3) = 17, a(4) = 71, a(15) = 1814380259, a(19) = 43758015399281, a(21) = 441080795274037, a(23) = 867081905243923.

			a(3) = 6 because 1/0!! + 1/1!! + 1/2!! + 1/3!! = 17/6.
a(15) = 593049600 because 1814380259/593049600 = 1/1 + 1/1 + 1/2 + 1/3 + 1/8 + 1/15 + 1/48 + 1/105 + 1/384 + 1/945 + 1/3840 + 1/10395 + 1/46080 + 1/135135 + 1/645120 + 1/2027025.

G. C. Greubel, Table of n, a(n) for n = 0..500
Eric W. Weisstein, Double Factorial. Gives formula for limit of series, which was independently derived by Max Alekseyev.

Cf. A006882 (n!!), A094007, A143280 (m(2)), A143382 (numerator).

[n le 0 select 1 else Denominator( 1 + (&+[ 1/(0 + (&*[k-2*j: j in [0..Floor((k-1)/2)]])) : k in [1..n]]) ): n in [0..25]]; // G. C. Greubel, Mar 28 2019

Table[Denominator[Sum[1/k!!, {k,0,n}]], {n,0,25}] (* G. C. Greubel, Mar 28 2019 *)

vector(25, n, n--; denominator(sum(k=0,n, 1/prod(j=0,floor((k-1)/2), (k - 2*j)) ))) \\ G. C. Greubel, Mar 28 2019

[denominator(sum(1/product((k-2*j) for j in (0..floor((k-1)/2))) for k in (0..n))) for n in (0..25)] # G. C. Greubel, Mar 28 2019

Denominators of Sum_{k=0..n} 1/k!! = Sum_{k=0..n} 1/A006882(k).

A138366 Count of post-period decimal digits up to which the rounded n-th convergent to exp(1) agrees with the exact value.

Original entry on oeis.org

0, 1, 0, 1, 2, 3, 4, 5, 5, 6, 7, 7, 8, 9, 10, 12, 12, 13, 14, 16, 15, 16, 19, 18, 20, 22, 22, 24, 25, 25, 26, 27, 28, 30, 32, 32, 32, 35, 36, 36, 39, 39, 41, 43, 43, 44, 46, 46, 48, 50, 50, 52, 52, 54, 56, 57, 58, 59, 61, 61, 63, 65, 64, 67, 69, 69, 71, 72, 73, 74, 77, 77, 79, 80, 81, 83

Offset: 1

Artur Jasinski, Mar 17 2008

This is a measure of the quality of the n-th convergent to E = A001113 if the convergent and the exact value are compared rounded to an increasing number of digits.
The sequence of rounded values of exp(1) is 3, 2.7, 2.72, 2.718, 2.7183, 2.71828, 2.718282, 2.7182818 etc, and the n-th convergent (provided by A007676 and A007677) is to be represented by its equivalent sequence.
a(n) represents the maximum number of post-period digits of the two sequences if compared at the same level of rounding. Counting only post-period digits (which is one less than the full number of decimal digits) is just a convention taken from A084407.

			For n=6, the 6th convergent is 106/39 = 2.7179487.., with a sequence of rounded representations 3, 2.7, 2.72, 2.718, 2.7179, 2.71795, 2.717949, etc.
Rounded to 1, 2, or 3 post-period decimal digits, this is the same as the rounded version of the exact E, but disagrees if both are rounded to 4 decimal digits, where 2.7183 <> 2.7179.
So a(6) = 3 (digits), the maximum rounding level of agreement.

Cf. A138335, A138336, A138337, A138339, A138343, A138367, A138369, A138370.

Definition and values replaced as defined via continued fractions by R. J. Mathar, Oct 01 2009

A368617 Decimal expansion of 878/323.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Jan 01 2024

Using the criterion of the minimum absolute difference, it is the closest rational approximation of e (cf. A001113) using integers below 1000 (see Maor).
The numerator and the denominator of 878/323 are palindromes in base 10 (cf. A002113).
It has period 144.

			2.7182662538699690402476780...

Eli Maor, e: The Story of a Number. Princeton, New Jersey: Princeton University Press (1994), p. 37.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 46.

Philip R. Brown, Approximations of e and Pi: an exploration, International Journal of Mathematical Education in Science and Technology, 48:sup1, S30-S39, 2017. See p. S32.
Index entries for sequences related to the number e.
Index entries for linear recurrences with constant coefficients, order 143.

Cf. A001113, A002113, A007676, A007677, A368653, A368654, A368656.

Cf. A368618, A368619, A368620, A368621.

Mathematica
```
Flatten[First[RealDigits[878/323,10]]]
```

Equals A368618(3)/A368619(3) = A368620(3)/A368621(3).

A368618 a(n) is the n-digit numerator of the fraction h/k with h and k coprime palindrome positive integers at which abs(h/k-e) is minimal.

Original entry on oeis.org

3, 11, 878, 2552, 38983, 167761, 4407044, 24988942, 882646288, 1385885831, 83034443038, 161356653161, 9051164611509, 24911822811942

Offset: 1

Stefano Spezia, Jan 01 2024

a(3) = 878 corresponds to the numerator of A368617.

			  n              fraction    approximated value
  -   -------------------    ------------------
  1                   3/1    3
  2                  11/4    2.75
  3               878/323    2.7182662538699...
  4              2552/939    2.7177848775292...
  5           38983/14341    2.7182902168607...
  6          167761/61716    2.7182740294251...
  7       4407044/1621261    2.7182816338640...
  8      24988942/9192919    2.7182815382143...
  9   882646288/324707423    2.7182818299783...
  ...

David A. Corneth, PARI program
Index entries for sequences related to the number e

Cf. A001113, A002113, A070252, A368617, A368619 (denominator), A368658.
Cf. A007676, A007677.
Cf. A364844 (similar for Pi), A368620, A368621.

a[1]=3; a[n_]:=Module[{minim = Infinity}, h = Select[Range[10^(n - 1), 10^n - 1], PalindromeQ]; lh = Length[h]; For[i = 1, i <= lh, i++, k = Select[Range[Floor[Part[h, i]/E], Ceiling[Part[h, i]/E]], PalindromeQ]; lk = Length[k]; For[j = 1, j <= lk, j++, If[(dist = Abs[Part[h, i]/Part[k, j] - E]) < minim && GCD[Part[h, i], Part[k, j]] == 1, minim = dist; hmin = Part[h, i]]]]; hmin]; Array[a,9]

PARI
```
\\ See PARI program in Links
```

a(10)-a(14) from David A. Corneth, Jan 03 2024

A079940 Greedy fractional multiples of 1/e: a(1)=1, Sum_{n>0} frac(a(n)/e) = 1.

Original entry on oeis.org

1, 3, 4, 11, 87, 193, 386, 579, 1457, 23225, 49171, 98342, 147513, 196684, 566827, 13580623, 28245729, 56491458, 84737187, 112982916, 438351041, 466596770, 494842499

Offset: 1

Benoit Cloitre and Paul D. Hanna, Jan 21 2003

The n-th greedy fractional multiple of x is the smallest integer m that does not cause Sum_{k=1..n} frac(m*x) to exceed unity; an infinite number of terms appear as the denominators of the convergents to the continued fraction of 1/e.

After a(20), there is only 109305220 - 297122396/e = ~1.06317354345346734...*10^-8 to work with.

			a(4) = 11 since frac(1x) + frac(3x) + frac(4x) + frac(11x) < 1, while frac(1x) + frac(3x) + frac(4x) + frac(k*x) > 1 for all k>4 and k<11.

K. Girstmair, On the Asymptotic Behavior of Dedekind Sums, J. Int. Seq. 17 (2014) # 14.7.7, example 2.

Cf. A007676 (numerators of convergents to e), A079934, A079939, A079941.

Digits := 100: a := []: s := 0: x := 1.0/exp(1.0): for n from 1 to 1000000 do: temp := evalf(s+frac(n*x)): if (temp<1.0) then a := [op(a),n]: print(n): s := s+evalf(frac(n*x)): fi: od: a;

a[1] = 1; a[n_] := a[n] = Block[{k = a[n - 1] + 1, ps = Plus @@ Table[ FractionalPart[ a[i]*E^-1], {i, n - 1}]}, While[ ps + FractionalPart[k*E^-1] > 1, k++ ]; a[n] = k]; Do[ Print[ a[n]], {n, 20}] (* Robert G. Wilson v, Nov 03 2004 *)

More terms from Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 30 2003
a(16)-a(20) from Robert G. Wilson v, Nov 03 2004
a(21)-a(23) from Sean A. Irvine, Aug 30 2025

A368619 a(n) is the n-digit denominator of the fraction h/k with h and k coprime palindrome positive integers at which abs(h/k-e) is minimal.

Original entry on oeis.org

1, 4, 323, 939, 14341, 61716, 1621261, 9192919, 324707423, 509838905, 30546664503, 59359795395, 3329737379233, 9164547454619

Offset: 1

Stefano Spezia, Jan 01 2024

a(3) = 323 corresponds to the denominator of A368617.

			  n              fraction    approximated value
  -   -------------------    ------------------
  1                   3/1    3
  2                  11/4    2.75
  3               878/323    2.7182662538699...
  4              2552/939    2.7177848775292...
  5           38983/14341    2.7182902168607...
  6          167761/61716    2.7182740294251...
  7       4407044/1621261    2.7182816338640...
  8      24988942/9192919    2.7182815382143...
  9   882646288/324707423    2.7182818299783...
  ...

David A. Corneth, PARI program
Index entries for sequences related to the number e

Cf. A001113, A002113, A070252, A368617, A368618 (numerator), A368658.
Cf. A007676, A007677.
Cf. A364845 (similar for Pi), A368620, A368621.

a[1]=1; a[n_]:=Module[{minim = Infinity}, h = Select[Range[10^(n - 1), 10^n - 1], PalindromeQ]; lh = Length[h]; For[i = 1, i <= lh, i++, k = Select[Range[Floor[Part[h, i]/E], Ceiling[Part[h, i]/E]], PalindromeQ]; lk = Length[k]; For[j = 1, j <= lk, j++, If[(dist = Abs[Part[h, i]/Part[k, j] - E]) < minim && GCD[Part[h, i], Part[k, j]] == 1, minim = dist; kmin = Part[k, j]]]]; kmin]; Array[a,9]

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a(10)-a(14) from David A. Corneth, Jan 03 2024

cp's OEIS Frontend

A136122 Integer log of (numerator of convergent to E / denominator of convergent to E) = A001414(A007676/A007677) = A001414(A007676)-A001414(A007677).

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A003417 Continued fraction for e.

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A007677 Denominators of convergents to e.

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A143382 Numerator of Sum_{k=0..n} 1/k!!.

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A143383 Denominator of Sum_{k=0..n} 1/k!!.

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A138366 Count of post-period decimal digits up to which the rounded n-th convergent to exp(1) agrees with the exact value.

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A368617 Decimal expansion of 878/323.

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