A007677 -id:A007677 - cp's OEIS frontend

A085368 Sum of numerators and denominators of convergents to 1/e.

3, 4, 11, 15, 26, 119, 145, 264, 1729, 1993, 3722, 31769, 35491, 67260, 708091, 775351, 1483442, 18576655, 20060097, 38636752, 560974625, 599611377, 1160586002, 19168987409, 20329573411, 39498560820, 731303668171, 770802228991, 1502105897162, 30812920172231

Offset: 1

Gary W. Adamson, Jun 26 2003

nonn

Through a(n) natural numbers 1,2,3...a(n), A007677(n-1) of those terms are members of the upper level Beatty sequence A000572; while A007676(n) of those terms are in the lower level Beatty sequence A006594.
Check: a(5) = 26, which has 7 (= A007677(4)) terms in A000572: 3, 7, 11, 14, 18, 22 and 26; while the remaining 19 (= A007676(5)) are members of the lower level Beatty sequence A006594.
A085368(n)/A007677(n-1) converge upon (1 + e), as n approaches infinity. Check: A085368(6)/A007677(5) = 119/32 = 3.71875... where (1 + e) =3.718281828... A085368(n)/A007676(n) converge upon (1 + 1/e). Check: A085368(5)/A007676(5) = 119/87 = 1.3678.., where (1 + 1/e) = 1.367879441... A006594 and A000572 form Beatty pairs, with floor n*(1 + e) being the generator for A000572(n) and floor n*(1 + 1/e) the generator for A006594(n).
The cutting sequence for y = (1/e)x is generated from the line starting at (0,0), passing through an array of squares, giving "1" to an intersection with a vertical line and "0" to an intersection with a horizontal line. The cutting sequence for y = (1/e)x is 0, then (terms 1 through 26): 1 1 0 1 1 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0. In this sequence, n's for 0's are all members of the upper Beatty pair: A000572 (check: n's for the 0's are 3, 7, 11, 14, 18, 22 and 26 (the 7 being A007677(4)); while 19 terms (19 = A007676(5)) are members of the lower Beatty pair A006594, being denoted by "1" and thus intersecting vertical lines.

			a(6) = 119 = 32 + 87 where 32/87 is the 6th convergent to 1/e: [2,1,2,1,1,4]= 32/87 = .367816...& 1/e = .3678794...
a(6) = 119 = 32 + 87 = A007677(5) + A007676(6).

Cf. A007677, A007676, A006594, A000572.

Convergents to 1/e are generated from the partial quotients of the continued fraction form of 1/e: [2, 1, 2, 1, 1, 4, 1, 1, 6...], where below each partial quotient, the first 9 convergents are 1/2, 1/3, 3/8...(i.e. 1/2 = [2], 1/3 = [2, 1], 3/8 = [2, 1, 2], etc;...then 4/11, 7/19, 32/87, 39/106, 71/193, 465/1264, where a(n) = sum of numerator and denominator of n-th convergent to 1/e with 1/2 = first convergent.

a(n) = A007676(n) + A007677(n-1) where A007676 = 2, 3, 8, 11, 19, 87...(numerators to convergents to e); and A007677 = 1, 1, 3, 4, 7, 32, 39, 71...(denominators of convergents to e).

More terms from Colin Barker, Mar 11 2014

A085369 Cutting sequence for 1/e.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0

Offset: 1

Gary W. Adamson, Jun 26 2003

nonn

Through any A085368(n) number of terms in the cutting sequence, A007677(n-1) of those terms are zeros and A007676(n) are ones. Check: A085368(5) = 26, the sequence being 3, 4, 11, 15, 26, ... (sum of numerators and denominators of convergents to 1/e). Then through n=26, A085369(n) is 1 1 0 1 1 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0, with 7 zeros and 19 ones, (7/19 being the 5th convergent to 1/e): 7/19 = [2, 1, 2, 1, 1]. Numerator and denominator sum = 26, with 7 zeros and 19 ones, with the zeros occupying positions n = 3, 7, 11, 14, 18, 22 and 26 (also being the first 7 terms of A000572). Positions of the cutting sequence occupied by ones (1, 2, 4, 5, 6, ...) are consecutive terms of the lower Beatty sequence A006594, being generated by floor(n*(1 + 1/e)).

			a(6) = 1, where 1's correspond to members of the lower Beatty pair A006594 which is generated from floor(n*(1 + 1/e)). Check: floor(5*(1 + 1/e)) = 6. All terms not in A006594 are 0's.
a(7) = 0, where 7 is not a member of A006594, but is a member of the upper Beatty pair sequence A000572 which has the generator floor(n*(e + 1)). Check: floor(2*(1 + e)) = 7.

Manfred R. Schroeder, "Fractals, Chaos, Power Laws", Freeman, 1996, p. 56.

Cf. A085368, A007677, A007676, A006594, A000572.

Given the line y = (1/e)x starting from (0, 0) and passing through an array of squares, a "1" denotes an intersection with a vertical line, while an "0" denotes an intersection with a horizontal line.

n for 0's are consecutive terms of upper Beatty pair terms A000572: 3, 7, 11, 14, 18, 22, 26, ..., while n's for all 1's are paired lower Beatty terms of A006594: 1, 2, 4, 5, 6, 8, ...

A102049 Indices of primes which are denominators of convergents to e.

Original entry on oeis.org

2, 4, 20, 2073, 688812, 23493068282804, 51287550456151700

Offset: 1

Jonathan Sondow, Dec 27 2004

The prime denominators of convergents to e form A094008 (so A000040(a(n)) = A094008(n)). Their positions in A007677 (denominators of convergents to e) form A094007, so a(n) = A000720(A007677(A094007(n))).

a(6)-a(7) computed using Kim Walisch's primecount program. - Giovanni Resta, Jun 03 2019

			a(1) = 2 because the first convergent to e with prime denominator is 8/3 and the index of 3 is 2, i.e., 3 is the 2nd prime.

E. B. Burger, Diophantine Olympics and World Champions: Polynomials and Primes Down Under, Amer. Math. Monthly, 107 (Nov. 2000), 822-829.
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.

Cf. A000040, A000720, A007677, A094007, A094008.

a(n) = A000720(A094008(n)).

a(6)-a(7) from Giovanni Resta, Jun 03 2019

A259589 Numerators of the other-side convergents to e.

Original entry on oeis.org

3, 5, 11, 19, 30, 106, 193, 299, 1457, 2721, 4178, 25946, 49171, 75117, 566827, 1084483, 1651310, 14665106, 28245729, 42910835, 438351041, 848456353, 1286807394, 14862109042, 28875761731, 43737870773, 563501581931, 1098127402131, 1661628984062

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. A259588, 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) = P(i).

A078977 Denominator of n-th convergent to e^(2/3).

Original entry on oeis.org

1, 1, 19, 134, 153, 287, 3023, 163529, 2619487, 2783016, 5402503, 105430573, 9494154073, 237459282398, 246953436471, 484412718869, 13810509564803, 1740608617884047, 59194503517622401, 60935112135506448

Offset: 1

Benoit Cloitre, Dec 19 2002

Cf. A069951, A065923, A001518, A007677, A078976.

Denominator[Convergents[E^(2/3),20]] (* Harvey P. Dale, Dec 01 2013 *)

a(n)=component(component(contfracpnqn(contfrac(exp(2/3),n)),1),2) \\ (Warning: this will give only a limited number of correct terms, depending on the precision used. - The Editors, Oct 13 2009. See A078976 for better code.)

Special cases : a(5k+1)=abs(A065923(3k)); a(5k+3)=abs(A065923(3k+2)) where A065923(n)=y(n, -3) where y(n, x)=sum (k=0, n, (n+k)!*(x/2)^k/((n-k)!*k!))

A108688 Smallest integer q >= 1 such that difference between q*e and the nearest integer is <= 1/n.

Original entry on oeis.org

1, 1, 3, 3, 3, 4, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 71, 71, 71, 71, 71, 71, 71, 71

Offset: 2

N. J. A. Sloane, Jun 18 2005

nonn

See A105960 for more information. If repeats are omitted we get A007677.

A233208 A measure of quality (the higher the better) for the approximation to e by rationals A022852(n)/n.

Original entry on oeis.org

3, 2, 6, 7, 2, 3, 35, 3, 2, 5, 10, 2, 2, 17, 4, 2, 4, 14, 2, 2, 11, 5, 2, 4, 23, 3, 2, 8, 5, 2, 3, 66, 3, 2, 7, 7, 2, 3, 76, 3, 2, 5, 8, 2, 3, 24, 4, 2, 5, 11, 2, 2, 14, 4, 2, 4, 17, 2, 2, 10, 5, 2, 3, 33, 3, 2, 8, 6, 2, 3, 502, 3, 2, 6, 7, 2, 3, 38, 3, 2, 5, 9, 2, 2, 18, 4, 2, 4, 13, 2, 2, 12, 5, 2, 4, 22

Offset: 1

Franz Vrabec, Dec 06 2013

nonn

a(n) is the greatest natural number such that abs( n*e-A022852(n) ) < 1/a(n). Trivially a(n)>=2. a(n)=2 iff n is in A191104 (easy proof).

			a(7) = 35 because floor(1/abs(7*e-19)) = floor(1/0.0279727...) = floor(35.749...) = 35.

Cf. A022852. For records see A233209, A007677.

a(n)=floor(1/abs(n*exp(1)-round(n*exp(1)))) \\ Ralf Stephan, Dec 13 2013

a(n) = floor( 1 / abs( n*e-A022852(n) ) ).

More terms from Ralf Stephan, Dec 13 2013

A233209 Records in A233208.

Original entry on oeis.org

3, 6, 7, 35, 66, 76, 502, 952, 1063, 9067, 17347, 18996, 199826, 384870, 415618, 5202154, 10069003, 10762714, 156217515, 303515018, 321895220, 5315503943, 10358584169, 10918147744, 202115644015, 394835308580, 414085365153, 8493248958168, 16625236738313, 17363808016482

Offset: 1

Franz Vrabec, Dec 06 2013

nonn

Cf. A233208. Positions of records is A007677.

m=0;for(n=1,10^6,t=floor(1/abs(n*exp(1)-round(n*exp(1))));if(t>m,m=t;print1(m,","))) \\ Ralf Stephan, Dec 13 2013

More terms from Ralf Stephan, Dec 13 2013

More terms from Jinyuan Wang, Mar 20 2020

A367329 The y-coordinate of the point where x + y = n, x & y are integers and x/y is as close as possible to e.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 19, 19, 19, 19, 20, 20

Offset: 1

Colin Linzer, Nov 14 2023

a(n) is nondecreasing; lim_{n->oo} a(n) = oo.

Swapping the x and y coordinate of the sequence does not yield the sequence defined as the point where x + y = n, x and y are integers and x/y is as close as possible to 1/e even when excluding terms that would lead to a division by 0.

			For n = 3, the possible points are (0,3), (1,2), (2,1), as any negative fraction would would be further from e than 0/3. The closest fraction to e out of these is 2/1 so a(3) = 1.

Cf. A001113 (e), A367328 (x-coordinate), A007677.

A114539 Number of correct decimal digits given by the n-th convergent to e.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 3, 4, 5, 5, 6, 8, 8, 9, 10, 10, 12, 13, 13, 15, 16, 16, 18, 19, 19, 21, 22, 22, 24, 26, 26, 27, 29, 29, 31, 32, 32, 34, 36, 36, 38, 39, 39, 41, 43, 43, 45, 47, 47, 48, 50, 50, 52, 54, 54, 56, 58, 58, 60, 62, 62, 64, 65, 66, 67, 69, 69, 71, 73, 73, 75, 77, 77

Offset: 0

Eric W. Weisstein, Dec 07 2005

Eric Weisstein's World of Mathematics, e Continued Fraction

Cf. A007676, A007677.

cp's OEIS Frontend

A085368 Sum of numerators and denominators of convergents to 1/e.

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A085369 Cutting sequence for 1/e.

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A102049 Indices of primes which are denominators of convergents to e.

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A259589 Numerators of the other-side convergents to e.

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A078977 Denominator of n-th convergent to e^(2/3).

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A108688 Smallest integer q >= 1 such that difference between q*e and the nearest integer is <= 1/n.

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A233208 A measure of quality (the higher the better) for the approximation to e by rationals A022852(n)/n.

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PARI

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A233209 Records in A233208.

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PARI

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A367329 The y-coordinate of the point where x + y = n, x & y are integers and x/y is as close as possible to e.

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Comments

Examples

Crossrefs

A114539 Number of correct decimal digits given by the n-th convergent to e.