A094008 -id:A094008 - cp's OEIS frontend

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

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).

A094007 Numbers k such that the denominator of the k-th convergent of the continued fraction expansion of e is prime.

Original entry on oeis.org

3, 5, 8, 14, 20, 35, 41, 65, 239, 269

Offset: 1

Jonathan Sondow, Apr 20 2004; corrected Apr 21 2004

a(n) is the position of A094008(n) in A007677 (denominators of convergents to e), so A007677(a(n)) = A094008(n). Also, A102049(n) is the position of A007677(a(n)) in A000040 (the prime numbers), so A000040(A102049(n)) = A007677(a(n)).

a(11) > 50000. - Lucas A. Brown, Apr 21 2021

			The convergents for e are 2, 3, 8/3, 11/4, 19/7, ... and so the 3rd convergent is the first one with prime denominator: a(1) = 3 and the 5th convergent is the 2nd one with prime denominator: a(2) = 5.

E. B. Burger, Diophantine Olympics and World Champions: Polynomials and Primes Down Under, Amer. Math. Monthly, 107 (Nov. 2000), 822-829.
Jonathan 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).
Jonathan Sondow, A geometric proof that e is irrational and a new measure of its irrationality, arXiv:0704.1282 [math.HO], 2007-2010.
Jonathan 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, A094008, A102049.

L = {}; cf = ContinuedFraction[E, 5000]; Do[ If[ PrimeQ[ Denominator[ FromContinuedFraction[ Take[ cf, n]] ]], AppendTo[L, n]], {n, Length[cf]}]; L (* Robert G. Wilson v, May 14 2004 *)

More terms from Robert G. Wilson v, May 14 2004

A086791 Primes found among the numerators of the continued fraction rational approximations to e.

Original entry on oeis.org

2, 3, 11, 19, 193, 49171, 1084483, 563501581931, 332993721039856822081, 3883282200001578119609988529770479452142437123001916048102414513139044082579

Offset: 1

Cino Hilliard, Aug 04 2003; corrected Jul 24 2004

			The first 8 rational approximations to e are 2/1, 3/1, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71. The numerators 2, 3, 11, 19, 193 are primes.

Joerg Arndt, Table of n, a(n) for n = 1..11
Cino Hilliard, Continued fractions rational approximation of numeric constants.

Cf. A002119, A086788, A094008.

\\ Continued fraction rational approximation of numeric constants f. m=steps.
cfracnumprime(m,f) = { default(realprecision,3000); cf = vector(m+10); x=f; for(n=0,m, i=floor(x); x=1/(x-i); cf[n+1] = i; ); for(m1=0,m, r=cf[m1+1]; forstep(n=m1,1,-1, r = 1/r; r+=cf[n]; ); numer=numerator(r); denom=denominator(r); if(ispseudoprime(numer),print1(numer,",")); ) }

default(realprecision,10^5);
cf=contfrac(exp(1));
n=0;
{ for(k=1, #cf,  \\ generate b-file
    pq = contfracpnqn( vector(k,j, cf[j]) );
    p = pq[1,1];  q = pq[2,1];
    if ( ispseudoprime(p), n+=1; print(n," ",p) );  \\ A086791
\\    if ( ispseudoprime(q), n+=1; print(n," ",q) );  \\ A094008
); }
/* Joerg Arndt, Apr 21 2013 */

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

cp's OEIS Frontend

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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A094007 Numbers k such that the denominator of the k-th convergent of the continued fraction expansion of e is prime.

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A086791 Primes found among the numerators of the continued fraction rational approximations to e.

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

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