A094007 -id:A094007 - 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).

A094008 Primes which are the denominators of convergents of the continued fraction expansion of e.

Original entry on oeis.org

3, 7, 71, 18089, 10391023, 781379079653017, 2111421691000680031, 1430286763442005122380663256416207

Offset: 1

Jonathan Sondow, Apr 20 2004

nonn

The position of a(n) in A000040 (the prime numbers) is A102049(n) = A000720(a(n)). - Jonathan Sondow, Dec 27 2004

The next term has 166 digits. [Harvey P. Dale, Aug 23 2011]

			a(1) = 3 because 3 is the first prime denominator of a convergent, 8/3, of the simple continued fraction for e

Joerg Arndt, Table of n, a(n) for n = 1..10
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. A094007.

See also A000040, A000720, A007677, A102049.

Block[{$MaxExtraPrecision=1000},Select[Denominator[Convergents[E,500]], PrimeQ]] (* Harvey P. Dale, Aug 23 2011 *)

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 */

a(n) = A007677(A094007(n)) = A000040(A102049(n)).

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

A259490 Numbers k such that the denominator of the n-th convergent of the continued fraction expansion of Pi is prime.

Original entry on oeis.org

2, 4, 9, 33, 595, 1127, 2003, 3611, 4356, 6926

Offset: 1

Jonathan Sondow and Robert G. Wilson v, Jun 28 2015

These are the k such that A002486(k+1) is prime. - Michel Marcus, Jul 15 2015

a(11) > 25000.

Cf. A094007, A002486.

$MaxExtraPrecision = 25000; lst = {}; cf = ContinuedFraction[Pi, 10000]; Do[ If[ PrimeQ[ Denominator[ FromContinuedFraction[ Take[ cf, n]] ]], AppendTo[lst, n]], {n, Length[cf]}]; lst
Position[Convergents[Pi,7000],?(PrimeQ[Denominator[#]]&)]//Flatten (* _Harvey P. Dale, Aug 12 2021 *)

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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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A094008 Primes which are the denominators of convergents of the continued fraction expansion of e.

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

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A259490 Numbers k such that the denominator of the n-th convergent of the continued fraction expansion of Pi is prime.

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