A061355 -id:A061355 - cp's OEIS frontend

A102470 Numbers n such that denominator of Sum_{k=0 to n} 1/k! is n!.

0, 1, 2, 4, 6, 8, 10, 16, 18, 20, 26, 28, 40, 46, 48, 58, 66, 68, 70, 80, 86, 96, 98, 118, 126, 130, 136, 146, 150, 170, 176, 178, 180, 188, 190, 206, 208, 210, 216, 230, 260, 266, 268, 278, 286, 288, 300, 306, 308, 326, 328, 338, 346, 358, 366, 370, 378, 380, 388

Offset: 1

Jonathan Sondow, Jan 14 2005

nonn

a(n) is even for n > 1, as Sum_{k=0 to n} 1/k! reduces to lower terms when n > 1 is odd.

			1/0! + 1/1! + 1/2! + 1/3! +1/4! = 65/24 and 24 = 4!, so 4 is a member. But 1/0! + 1/1! + 1/2! + 1/3! = 8/3 and 3 < 3!, so 3 is not a member.

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.
Index entries for sequences related to factorial numbers

For n > 0, n is a member <=> A093101(n) = 1 <=> A061355(n) = n! <=> A061355(n) = A002034(A061355(n))! <=> A061354(n) = 1+n+n(n-1)+n(n-1)(n-2)+...+n!. See also A102471.

fQ[n_] := (Denominator[Sum[1/k!, {k, 0, n}]] == n!); Select[ Range[0, 389], fQ[ # ] &] (* Robert G. Wilson v, Jan 15 2005 *)

a(n) = 2*A102471(n-1) for n > 1.

More terms from Robert G. Wilson v, Jan 15 2005

A102471 Numbers n such that the denominator of Sum_{k=0 to 2n} 1/k! is (2n)!.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 9, 10, 13, 14, 20, 23, 24, 29, 33, 34, 35, 40, 43, 48, 49, 59, 63, 65, 68, 73, 75, 85, 88, 89, 90, 94, 95, 103, 104, 105, 108, 115, 130, 133, 134, 139, 143, 144, 150, 153, 154, 163, 164, 169, 173, 179, 183, 185, 189, 190, 194, 195, 198, 199, 204

Offset: 1

Jonathan Sondow, Jan 14 2005

nonn

n is a member <=> A093101(2n) = 1 <=> A061355(2n) = (2n)! <=> A061355(2n) = A002034(A061355(2n))!.

			Sum_{k=0 to 6} 1/k! = 1957/720 and 720 = 6! = (2*3)!, so 3 is a member. But Sum_{k=0 to 12} 1/k! = 260412269/95800320 and 95800320 < 12! = (2*6)!, so 6 is not a member.

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.
Index entries for sequences related to factorial numbers

Cf. A102470, A093101, A061355, A002034.

fQ[n_] := (Denominator[Sum[1/k!, {k, 0, 2n}]] == (2n)!); Select[ Range[0, 204], fQ[ # ] &] (* Robert G. Wilson v, Jan 15 2005 *)

a(n) = A102470(n+1)/2 for n > 0.

More terms from Robert G. Wilson v, Jan 15 2005

A195326 Numerators of fractions leading to e - 1/e (A174548).

Original entry on oeis.org

0, 2, 2, 7, 7, 47, 47, 5923, 5923, 426457, 426457, 15636757, 15636757, 7318002277, 7318002277, 1536780478171, 1536780478171, 603180793741, 603180793741, 142957467201379447, 142957467201379447

Offset: 0

Paul Curtz, Oct 12 2011

The sequence of approximations of exp(1) obtained by truncating the Taylor series of exp(x) after n terms is A061354(n)/A061355(n) = 1, 2, 5/2, 8/3, 65/24, ...
A Taylor series of exp(-1) is 1, 0, 1/2, 1/3, 3/8, ... and (apart from the first 2 terms) given by A000255(n)/A001048(n). Subtracting both sequences term by term we obtain a series for exp(1) - exp(-1) = 0, 2, 2, 7/3, 7/3, 47/20, 47/20, 5923/2520, 5923/2520, 426457/181440, 426457/181440, ... which defines the numerators here.
Each second of the denominators (that is 3, 2520, 19958400, ...) is found in A085990 (where each third term, that is 60, 19958400, ...) is to be omitted.
This numerator sequence here is basically obtained by doubling entries of A051397, A009628, A087208, or A186763, caused by the standard associations between cosh(x), sinh(x) and exp(x).

			a(0) =  1  -  1;
a(1) =  2  -  0;
a(2) = 5/2 - 1/2.

Cf. A001113, A068985, A197222, A197223.

taylExp1 := proc(n)
        add(1/j!,j=0..n) ;
end proc:
A000255 := proc(n)
        if n <=1 then
                1;
        else
                n*procname(n-1)+(n-1)*procname(n-2) ;
        end if;
end proc:
A001048 := proc(n)
        n!+(n-1)! ;
end proc:
A195326 := proc(n)
        if n = 0 then
                0;
        elif n =1 then
                2;
        else
                taylExp1(n) -A000255(n-2)/A001048(n-1);
        end if;
          numer(%);
end proc:
seq(A195326(n),n=0..20) ; # R. J. Mathar, Oct 14 2011

Material meant to be placed in other sequences removed by R. J. Mathar, Oct 14 2011

A235214 Decimal expansion of exp(exp(1) + 1).

Original entry on oeis.org

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

Offset: 2

Richard R. Forberg, Jan 04 2014

May also be written as e*(e^e).

			41.19355567471612356318828...

Ovidiu Furdui, Solution to problem 91.H, Mathematical Gazette 92(523), 2008, pp. 174-175.

Cf. A005493, A234473 (e^e/e), A073226 (e^e), A001113 (e).

Cf. A061354, A061355.

RealDigits[E^(E + 1), 10, 100][[1]] (* Alonso del Arte, Jan 05 2014 *)

exp(exp(1)+1) \\ Charles R Greathouse IV, Jan 14 2014

Equals Sum_{n>=0} A005493(n)/n!.
Equals 2*lim_{n->oo} n*(exp(Sum_{k=0..n} 1/k!) - ((1+1/n)^n)^e). See the Mathematical Gazette link. - Michel Marcus, Oct 24 2017
Equals Sum_{k>=1} e^k/(k-1)!. - Amiram Eldar, Jul 28 2020

More terms from Rick L. Shepherd, Jan 25 2014

A354378 a(n) is the denominator of Sum_{k=0..n} (-1)^k / (2*k)!.

Original entry on oeis.org

1, 2, 24, 720, 8064, 3628800, 479001600, 87178291200, 20922789888000, 1280474741145600, 2432902008176640000, 1124000727777607680000, 620448401733239439360000, 403291461126605635584000000, 60977668922342772100300800000, 1569543549184562477137920000000

Offset: 0

Ilya Gutkovskiy, May 24 2022

			1, 1/2, 13/24, 389/720, 4357/8064, 1960649/3628800, 258805669/479001600, ...

Cf. A010050, A049470, A053556, A061355, A143383, A354138 (numerators), A354331, A354333, A354335.

Table[Sum[(-1)^k/(2 k)!, {k, 0, n}], {n, 0, 15}] // Denominator
nmax = 15; CoefficientList[Series[Cos[Sqrt[x]]/(1 - x), {x, 0, nmax}], x] // Denominator

a(n) = denominator(sum(k=0, n, (-1)^k/(2*k)!)); \\ Michel Marcus, May 24 2022

Denominators of coefficients in expansion of cos(sqrt(x)) / (1 - x).

A102584 a(n) = 1/2 times the cancellation factor in reducing Sum_{k=0 to 2n+1} 1/k! to lowest terms.

Original entry on oeis.org

1, 1, 10, 5, 4, 1, 2, 65, 2000, 1, 26, 247, 20, 5, 2, 19, 8, 115, 10, 23, 52, 31, 10, 65, 416, 37, 2, 25, 20, 1, 38, 1, 40, 325, 1406, 37, 676, 65, 10, 63829, 368, 1, 230, 5, 4, 1, 26, 5, 40, 247, 26, 43, 3100, 9785, 2, 1, 256, 5, 2050, 13, 388, 1, 4810, 1495, 8, 23, 254, 5

Offset: 1

Jonathan Sondow, Jan 22 2005

nonn

The denominator of Sum_{k=0 to m} 1/k! is m!/d, where d = A093101(m). If m = 2n+1 > 1, then d is even and a(n) = d/2.

			1/0! + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + 1/6! + 1/7! = 13700/5040 = (20*685)/(20*252) and 7 = 2*3+1, so a(3) = 20/2 = 10.

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, arXiv:0709.0671 [math.NT], 2007-2009; Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.
Index entries for sequences related to factorial numbers.

a(n) = A093101(2n+1)/2 = (2n+1)!/(2*A061355(2n+1)).

A129978 Numbers k such that A120265(k) = numerator(Sum_{j=1..k} 1/j!) is a prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 12, 16, 19, 21, 22, 25, 41, 114, 181, 236, 2003, 6138

Offset: 1

Alexander Adamchuk, Jun 13 2007

Corresponding primes are A120265(a(n)) = {3, 5, 41, 103, 1237, 433, 164611949, 35951249665217, 52255141388393, 43894318766250120011, 386270005143001056097, 53952693026046706215979, 1249584099900912571604389306768231303904375213027, ...}.

a(17) > 1000; A120265(1000) ~ 2.9*10^2564 = (e-1)*A061355(1000). - M. F. Hasler, Jun 18 2007

Cf. A120265, A061354, A061355.

Do[ f=Numerator[ Sum[ 1/k!, {k,1,n} ] ]; If[ PrimeQ[f], Print[{n,f}] ], {n,1,236} ]
Flatten[Position[Numerator[Accumulate[1/Range[2150]!]],?PrimeQ]] (* _Harvey P. Dale, May 03 2013 *)

my(t=0); for( n=1,1000, if( ispseudoprime( numerator( t+=1/n!)), print1( n", " ))) \\ M. F. Hasler, Jun 18 2007

Edited by M. F. Hasler, Jun 18 2007
a(17) from Alexander Adamchuk, May 02 2010
a(18) from Michael S. Branicky, Sep 24 2024

A196080 Numerators of the sum of the n-th partial sums of the expansions of e and 1/e.

Original entry on oeis.org

2, 2, 3, 3, 37, 37, 1111, 1111, 6913, 6913, 799933, 799933, 739138093, 739138093, 44841044309, 44841044309, 32285551902481, 32285551902481, 9879378882159187, 9879378882159187, 1251387991740163687

Offset: 0

Paul Curtz, Sep 27 2011

The n-th partial sums of the Taylor expansion of E are f(n) = A061354(n)/A061355(n) = 1, 2, 5/2, 8/3, 65/24, 163/60,.. .
The partial sums of an expansion of 1/e are essentially A000255(n-2)/A001048(n-1) preceded by 1 and 0, namely  g(n)= 1, 0, 1/2, 1/3, 3/8, 11/30, 53/144, 103/280, 2119/5760,... (Jolley's partial sums of 1/E in A068985 is the bisection 0, 1/3, 11/30, 103/280, 16687/45360,... of g(n).)
The current sequence are the numerators of f(n)+g(n), converging to E+1/E, namely 2, 2, 3, 3, 37/12, 37/12, 1111/360, 1111/360, 6913/2240 = 62217/21060, 6913/2240 = 62217/21060, 799933/259200 = 5599531/1814400,... The unreduced fractions are apparently given by duplicated A051396(n+1)/A002674(n).

			a(0)=1+1, a(1)=2+0, a(2)=(5+1)/2, a(3)=(8+1)/3.

Cf. A001113, A068985, A137204 (e+1/e).

a[n_] := (E*Gamma[n+1, 1] + (1/E)*Gamma[n+1, -1])/n! // FullSimplify // Numerator; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Aug 02 2012 *)

Redefined by reduced fractions. - R. J. Mathar, Jul 02 2012

A233044 Pairs p, q for those partial sums p/q of the series e = sum_{n>=0} 1/n! that are not convergents to e.

Original entry on oeis.org

1, 1, 5, 2, 65, 24, 163, 60, 1957, 720, 685, 252, 109601, 40320, 98641, 36288, 9864101, 3628800, 13563139, 4989600, 260412269, 95800320, 8463398743, 3113510400, 47395032961, 17435658240, 888656868019, 326918592000

Offset: 1

Jonathan Sondow, Dec 07 2013

nonn

Sondow (2006) conjectured that 2/1 and 8/3 are the only partial sums of the Taylor series for e that are also convergents to the simple continued fraction for e. Sondow and Schalm (2008, 2010) proved partial results toward the conjecture. Berndt, Kim, and Zaharescu (2012) proved it in full.

			1/1, 5/2, 65/24, 163/60, 1957/720, 685/252, 109601/40320, 98641/36288, 9864101/3628800, 13563139/4989600, 260412269/95800320, 8463398743/3113510400, 47395032961/17435658240, 888656868019/326918592000

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), part I, in Tapas in Experimental Mathematics, T. Amdeberhan and V. H. Moll, eds., Contemp. Math., vol. 457, American Mathematical Society, Providence, RI, 2008, pp. 273-284.

B. Berndt, S. Kim, and A. Zaharescu, Diophantine approximation of the exponential function and Sondow's conjecture, abstract 2012.
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 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), part II, in Gems in Experimental Mathematics, T. Amdeberhan, L. A. Medina, V. H. Moll, eds., Contemp. Math., vol. 517, American Mathematical Society, Providence, RI, 2010, pp. 349-363.

Cf. A061354, A061355, A007676, A007677.

a(2n-1)/a(2n) = A061354(k)/A061355(k) for some k <> 1 and 3.

a(2n-1)/a(2n) <> A007676(k)/A007677(k) for all k.

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A102470 Numbers n such that denominator of Sum_{k=0 to n} 1/k! is n!.

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A102471 Numbers n such that the denominator of Sum_{k=0 to 2n} 1/k! is (2n)!.

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A195326 Numerators of fractions leading to e - 1/e (A174548).

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A235214 Decimal expansion of exp(exp(1) + 1).

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A354378 a(n) is the denominator of Sum_{k=0..n} (-1)^k / (2*k)!.

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A102584 a(n) = 1/2 times the cancellation factor in reducing Sum_{k=0 to 2n+1} 1/k! to lowest terms.

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A129978 Numbers k such that A120265(k) = numerator(Sum_{j=1..k} 1/j!) is a prime.

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A196080 Numerators of the sum of the n-th partial sums of the expansions of e and 1/e.