A007677 -id:A007677 - cp's OEIS frontend

A368620 a(n) is the n-digit numerator of the fraction h/k with h and k coprime positive integers at which abs(h/k-e) is minimal.

3, 87, 878, 2721, 49171, 566827, 9242691, 28245729, 848456353

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                 87/32    2.71875
  3               878/323    2.7182662538699...
  4             2721/1001    2.7182817182817...
  5           49171/18089    2.7182818287356...
  6         566827/208524    2.7182818284705...
  7       9242691/3400196    2.7182818284593...
  8     28245729/10391023    2.7182818284590...
  ...

Index entries for sequences related to the number e

Cf. A001113, A368617, A368621 (denominator), A368659.

Cf. A007676, A007677, A368618, A368619.

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

A368621 a(n) is the n-digit denominator of the fraction h/k with h and k coprime positive integers at which abs(h/k-e) is minimal.

Original entry on oeis.org

1, 32, 323, 1001, 18089, 208524, 3400196, 10391023, 312129649

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                 87/32    2.71875
  3               878/323    2.7182662538699...
  4             2721/1001    2.7182817182817...
  5           49171/18089    2.7182818287356...
  6         566827/208524    2.7182818284705...
  7       9242691/3400196    2.7182818284593...
  8     28245729/10391023    2.7182818284590...
  ...

Index entries for sequences related to the number e

Cf. A001113, A368617, A368620 (numerator), A368659.

Cf. A007676, A007677, A368618, A368619.

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

A113874 a(3n) = a(3n-1) + a(3n-2), a(3n+1) = 2n*a(3n) + a(3n-1), a(3n+2) = a(3n+1) + a(3n).

Original entry on oeis.org

1, 0, 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

Offset: 0

N. J. A. Sloane, Jan 27 2006

A113873(n)/a(n) -> e.

Without the first two terms, same as A007677 (denominators of convergents to e). - Jonathan Sondow, Aug 16 2006

T. D. Noe, Table of n, a(n) for n = 0..201
H. Cohn, A short proof of the simple continued fraction expansion of e, Amer. Math. Monthly, 113 (No. 1, 2006), 57-62.
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641.
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.

Cf. A113873.

a[0]:=1: a[1]:=0: a[2]:=1: for n from 3 to 34 do if n mod 3 = 0 then a[n]:=a[n-1]+a[n-2] elif n mod 3 = 1 then a[n]:=2*(n-1)*a[n-1]/3+a[n-2] else a[n]:=a[n-1]+a[n-2] fi: od: seq(a[n],n=0..34); # Emeric Deutsch, Jan 28 2006

a[0] = 1; a[1] = 0; a[n_] := a[n] = Switch[ Mod[n, 3], 0, a[n - 1] + a[n - 2], 1, 2(n - 1)/3*a[n - 1] + a[n - 2], 2, a[n - 1] + a[n - 2]]; a /@ Range[0, 30]
Join[{1,0},Denominator[Convergents[E,30]]] (* Harvey P. Dale, Aug 09 2014 *)

More terms from Robert G. Wilson v and Emeric Deutsch, Jan 28 2006

A259588 Denominators of the other-side convergents to e.

Original entry on oeis.org

1, 2, 4, 7, 11, 39, 71, 110, 536, 1001, 1537, 9545, 18089, 27634, 208524, 398959, 607483, 5394991, 10391023, 15786014, 161260336, 312129649, 473389985, 5467464369, 10622799089, 16090263458, 207300647060, 403978495031, 611279142091, 8690849042711

Offset: 0

Clark Kimberling, Jul 17 2015

Suppose that a positive irrational number r has continued fraction [a(0), a(1), ... ]. Define sequences p(i), q(i), P(i), Q(i) from the numerators and denominators of finite continued fractions as follows:
p(i)/q(i) = [a(0), a(1), ... a(i)] and P(i)/Q(i) = [a(0), a(1), ..., a(i) + 1]. The fractions p(i)/q(i) are the convergents to r, and the fractions P(i)/Q(i) are introduced here as the "other-side convergents" to
r, because p(2k)/q(2k) < r < P(2k)/Q(2k) and P(2k+1)/Q(2k+1) < r < p(2k+1)/q(2k+1), for k >= 0.
Closeness of P(i)/Q(i) to r is indicated by
|r - P(i)/Q(i)| < |p(i)/q(i) - P(i)/Q(i)| = 1/(q(i)Q(i)), for i >= 0.

			For r = e, the first 13 other-side convergents are 3/1, 5/2, 11/4, 19/7, 30/11, 106/39, 193/71, 299/110, 1457/536, 2721/1001, 4178/1537, 25946/9545, 49171/18089.
A comparison of convergents with other-side convergents:
i    p(i)/q(i)      P(i)/Q(i)  p(i)Q(i)-P(i)q(i)
0       2/1   < e <   3/1              -1
1       3/1   > e >   5/2               1
2       8/3   < e <   11/4             -1
3      11/4   > e >   19/7              1
4      19/7   < e <   30/11            -1
5      87/32  > e >  106/39             1

Cf. A259589, A007676, A007677.

r = E; a[i_] := Take[ContinuedFraction[r, 35], i];
b[i_] := ReplacePart[a[i], i -> Last[a[i]] + 1];
t = Table[FromContinuedFraction[b[i]], {i, 1, 35}]
u = Denominator[t]  (* A259588 *)
v = Numerator[t]    (* A259589 *)

p(i)*Q(i) - P(i)*q(i) = (-1)^(i+1), for i >= 0, where a(i) = Q(i).

A340737 Numerators of a sequence of fractions converging to e.

Original entry on oeis.org

3, 5, 19, 49, 193, 685, 2721, 12341, 49171, 271801, 1084483, 7073725, 28245729, 212385209, 848456353, 7226001865, 28875761731, 274743964621, 1098127402131, 11544775603241, 46150226651233, 531276670190245, 2124008553358849, 26573182030311229, 106246577894593683, 1435390805853694145

Offset: 1

Gary Detlefs, Jan 18 2021

This sequence is a subset of the numerators of a sequence of fractions converging to e which was obtained by the use of a program which searched for a fraction having a closer value to e than the preceding one. The initial terms of this sequence were 3/1, 5/2, 8/3, 11/4, 19/7, 49/18, 68/25, 87/32, 106/39, 193/71, 685/252, 878/323, 1071/394, 1264/465, 1457/536, 2721/1001, 12341/4540. The subset of the numerators filtered out of this sequence are a(1)..a(8).

The convergence is conjectured.

			Sequence of fractions begins 3/1, 5/2, 19/7, 49/18, 193/71, 685/252, 2721/1001, 12341/4540, ...

Denominators are listed in A340738.

Cf. A007676/A007677.

e:=proc(a,b,n)option remember; e(a,b,1):=a; e(a,b,2):=b; if n>2 and n mod 2 =1 then 2*e(a,b,n-1)+n*e(a,b,n-2) else if n>3 and n mod 2 = 0 then (n+2)*e(a,b,n-1)/2 -(e(a,b,n-2)+(n-2)*e(a,b,n-3)/2) fi fi end
seq(e(3,5,n), n = 1..20)
# code to print the sequence of fractions and error
for n from 1` to 20 do print(e(3,5,n)/e(1,2,n), evalf(exp(1)-e(3,5,n)/e(1,2,n)) od

a[1] = 3; a[2] = 5; a[n_] := a[n] = If[EvenQ[n], (n + 2)*a[n - 1]/2 - (a[n - 2] + (n - 2)*a[n - 3]/2), 2*a[n - 1] + n*a[n - 2]]; Array[a, 20] (* Amiram Eldar, Jan 18 2021 *)

a(1) = 3, a(2) = 5; for n > 2, a(n) = (n+2)*a(n-1)/2 - a(n-2) - (n-2)*a(n-3)/2 if n is even, 2*a(n-1) + n*a(n-2) otherwise.

A340738 Denominator of a sequence of fractions converging to e.

Original entry on oeis.org

1, 2, 7, 18, 71, 252, 1001, 4540, 18089, 99990, 398959, 2602278, 10391023, 78132152, 312129649, 2658297528, 10622799089, 101072656170, 403978495031, 4247085597370, 16977719590391, 195445764537012, 781379079653017, 9775727355457908, 39085931702241241, 528050767520083262, 2111421691000680031

Offset: 1

Gary Detlefs, Jan 18 2021

This sequence is a subset of the numerators of a sequence of fractions converging to e which was obtained by the use of a program which searched for a fraction having a closer value to e than the preceding one. The initial terms of this sequence were 3/1, 5/2, 8/3, 11/4, 19/7, 49/18, 68/25, 87/32, 106/39, 193/71, 685/252, 878/323, 1071/394, 1264/465, 1457/536, 2721/1001, 12341/4540. The subset of the denominators filtered out of this sequence are a(1)..a(8).

The convergence is conjectured.

			Sequence of fractions begins 3/1, 5/2, 19/7, 49/18, 193/71, 685/252, 2721/1001, 12341/4540, ...

Numerators are listed in A340737.

Cf. A007676/A007677.

e:=proc(a,b,n)option remember; e(a,b,1):=a; e(a,b,2):=b; if n>2 and n mod 2 =1 then 2*e(a,b,n-1)+n*e(a,b,n-2) else if n>3 and n mod 2 = 0 then (n+2)*e(a,b,n-1)/2 -(e(a,b,n-2)+(n-2)*e(a,b,n-3)/2) fi fi end
seq(e(1,2,n), n = 1..20)
# code to print the sequence of fractions and error
for n from 1` to 20 do print(e(3,5,n)/e(1,2,n), evalf(exp(1)-e(3,5,n)/e(1,2,n)) od

a[1] = 1; a[2] = 2; a[n_] := a[n] = If[EvenQ[n], (n + 2)*a[n - 1]/2 - (a[n - 2] + (n - 2)*a[n - 3]/2), 2*a[n - 1] + n*a[n - 2]]; Array[a, 20] (* Amiram Eldar, Jan 18 2021 *)

a(1) = 1, a(2) = 2; for n > 2, a(n) = (n+2)*a(n-1)/2 - a(n-2) - (n-2)*a(n-3)/2 if n is even, 2*a(n-1) + n*a(n-2) otherwise.

A368653 Decimal expansion of 58291/21444.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Jan 02 2024

It is a rational approximation of e having an error less than 0.0003% provided by Charles Hermite in 1874 (see Hermite and Maor), where the error is calculated by abs(58291/21444-e)/e and expressed in percent.

Periodic with a period length of 893. - Ray Chandler, Jan 19 2024

			2.7182894982279425480320835664987875396381272...

Eli Maor, e: The Story of a Number. Princeton, New Jersey: Princeton University Press (1994), p. 189.

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.
Charles Hermite, Sur la fonction exponentielle, Gauthier-Villars, Paris, 1874.
Index entries for sequences related to the number e
Index entries for linear recurrences with constant coefficients, order 893.

Cf. A001113, A007676, A007677, A368617, A368654, A368656.

Flatten[First[RealDigits[58291/21444,10,100]]]

A368654 Decimal expansion of 158452/21444.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Jan 02 2024

It is a rational approximation of e^2 (A072334) provided by Charles Hermite in 1874 (see Hermite and Maor).

Periodic with a period length of 893. - Ray Chandler, Jan 19 2024

			7.38910650997948144002984517813840701361686...

Eli Maor, e: The Story of a Number. Princeton, New Jersey: Princeton University Press (1994), p. 189.

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.
Charles Hermite, Sur la fonction exponentielle, Gauthier-Villars, Paris, 1874.
Index entries for sequences related to the number e
Index entries for linear recurrences with constant coefficients, order 893.

Cf. A001113, A007676, A007677, A072334, A368617, A368653, A368656.

Flatten[First[RealDigits[158452/21444,10,100]]]

A368656 Decimal expansion of 271801/99990.

Original entry on oeis.org

2, 7, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8

Offset: 1

Stefano Spezia, Jan 02 2024

Periodic with a period length of 4. - Ray Chandler, Jan 19 2024

			2.718281828182818281828182818281828182818281828...

Eli Maor, e: The Story of a Number. Princeton, New Jersey: Princeton University Press (1994), p. 37.

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. S30.
Index entries for sequences related to the number e
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A001113, A007676, A007677, A368617, A368653, A368654.

Flatten[First[RealDigits[271801/99990,10,100]]]

G.f.: x*(-x^5 - 8*x^3 - x^2 - 7*x - 2)/(x^4 - 1). - Chai Wah Wu, Jan 04 2024

A036413 Values of k for which there are no empty intervals when fractional_part(m*e) for m = 1, ..., k is plotted along [ 0, 1 ] subdivided into k equal regions.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 32, 35, 39, 71, 465, 536, 1001, 8544, 9545, 18089, 190435, 208524, 398959

Offset: 1

Eric W. Weisstein

The sequence is infinite since it contains all denominators of continued fraction convergents of e. The next such denominator is 190435, but it has yet to be proved that there are no other terms before it. - Ivan Neretin, Jan 23 2018
a(17) = 190435 verified as the next term. - Sean A. Irvine, Oct 31 2020
No other terms < 10^6. - Eric W. Weisstein, Apr 21 2024

Eric Weisstein's World of Mathematics, E.
Eric Weisstein's World of Mathematics, Equidistributed Sequence.

Cf. A036412.

Cf. A007677 (denominators of convergents to e).

With[{f = FractionalPart[E Range[1000]]}, Position[Table[Count[BinCounts[Take[f, n], {0., 1, 1/n}], 0], {n, Length[f]}], 0]] // Flatten (* Eric W. Weisstein, Apr 27 2024 *)

a(17) from Sean A. Irvine, Oct 31 2020

a(18)-a(19) from Eric W. Weisstein, Apr 19 2024

cp's OEIS Frontend

A368620 a(n) is the n-digit numerator of the fraction h/k with h and k coprime positive integers at which abs(h/k-e) is minimal.

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A368621 a(n) is the n-digit denominator of the fraction h/k with h and k coprime positive integers at which abs(h/k-e) is minimal.

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A113874 a(3n) = a(3n-1) + a(3n-2), a(3n+1) = 2n*a(3n) + a(3n-1), a(3n+2) = a(3n+1) + a(3n).

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A259588 Denominators of the other-side convergents to e.

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A340737 Numerators of a sequence of fractions converging to e.

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A340738 Denominator of a sequence of fractions converging to e.

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A368653 Decimal expansion of 58291/21444.

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A368654 Decimal expansion of 158452/21444.

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