A061354 -id:A061354 - cp's OEIS frontend

A330030 Least k such that Sum_{i=0..n} k^n / i! is a positive integer.

1, 1, 2, 3, 6, 30, 30, 42, 210, 42, 210, 2310, 2310, 30030, 30030, 30030, 30030, 39270, 510510, 1939938, 9699690, 9699690, 9699690, 17160990, 223092870, 903210, 223092870, 223092870, 223092870, 6469693230, 6469693230, 200560490130, 200560490130, 10555815270, 200560490130

Offset: 0

Jinyuan Wang, Mar 07 2020

nonn

Least k > 0 such that k^n/A061355(n) is an integer.

			For n = 7, the denominator of Sum_{i=0..7} 1/i! is 252 = 2^2*3^2*7, so a(7) = 2*3*7 = 42.

Jinyuan Wang, Table of n, a(n) for n = 0..500

Cf. A000142, A007947, A061354, A061355, A332734, A333196.

a(n) = factorback(factorint(denominator(sum(i=2, n, 1/i!)))[, 1]);

a(n) = A007947(A061355(n)).

A354302 a(n) is the numerator of Sum_{k=0..n} 1 / (k!)^2.

Original entry on oeis.org

1, 2, 9, 41, 1313, 5471, 1181737, 28952557, 1235309099, 150090055529, 30018011105801, 201787741322329, 523033825507476769, 44196358255381786981, 5774990812036553498851, 1949059399062336805862213, 997918412319916444601453057, 3697415655903280160125896583

Offset: 0

Ilya Gutkovskiy, May 23 2022

			1, 2, 9/4, 41/18, 1313/576, 5471/2400, 1181737/518400, 28952557/12700800, 1235309099/541900800, ...

Cf. A001044, A006040, A053557, A061354, A070910, A103816, A120265, A143382, A354303 (denominators), A354304.

Table[Sum[1/(k!)^2, {k, 0, n}], {n, 0, 17}] // Numerator
nmax = 17; CoefficientList[Series[BesselI[0, 2 Sqrt[x]]/(1 - x), {x, 0, nmax}], x] // Numerator

Numerators of coefficients in expansion of BesselI(0,2*sqrt(x)) / (1 - x).

A354304 a(n) is the numerator of Sum_{k=0..n} (-1)^k / (k!)^2.

Original entry on oeis.org

1, 0, 1, 2, 43, 403, 23213, 118483, 51997111, 1842647621, 327581799289, 8918414485643, 4670006130663971, 361730891537680087, 130890931830249779173, 427294615628884602769, 6534075316966068976316143, 885163015595247156635327497, 41526561745210509140249210357

Offset: 0

Ilya Gutkovskiy, May 23 2022

			1, 0, 1/4, 2/9, 43/192, 403/1800, 23213/103680, 118483/529200, 51997111/232243200, 1842647621/8230118400, ...

Cf. A001044, A053557, A061354, A073701, A091681, A103816, A120265, A143382, A354302, A354305 (denominators).

Table[Sum[(-1)^k/(k!)^2, {k, 0, n}], {n, 0, 18}] // Numerator
nmax = 18; CoefficientList[Series[BesselJ[0, 2 Sqrt[x]]/(1 - x), {x, 0, nmax}], x] // Numerator

Numerators of coefficients in expansion of BesselJ(0,2*sqrt(x)) / (1 - x).

A102468 a(n)! is the smallest factorial divisible by the numerator of Sum_{k=0...n} 1/k!, with a(0) = 1.

Original entry on oeis.org

1, 2, 5, 4, 13, 163, 103, 137, 863, 98641, 10687, 31469, 1540901, 522787, 5441, 226871807, 13619, 1276861, 414026539, 2124467, 12670743557, 838025081381, 44659157, 323895443, 337310723185584470837549, 54352957

Offset: 0

Jonathan Sondow, Jan 09 2005

nonn

It appears that a(n) = A102469(n) (largest prime factor of the same numerator) except when n = 3. The smallest factorial divisible by the corresponding denominator is n!. Omitting the 0th term in the sum, it appears that the Kempner number (A002034) and the largest prime factor, of the numerator of Sum_{k=1...n} 1/k! are both equal to A096058(n).

The Mathematica program given below was used to generate the sequence. If the numerator of Sum_{k=0...n}(1/k!) is squarefree, the program prints the value of the numerator's largest prime factor, which must equal a(n). Otherwise, the program prints the complete factorization of the numerator so a(n) can be determined by inspection. - Ryan Propper, Jul 31 2005

			Sum_{k=0...3} 1/k! = 8/3 and 4! is the smallest factorial divisible by 8, so a(3) = 4.

A. J. Kempner, Miscellanea, Amer. Math. Monthly, 25 (1918), 201-210 [ See Section II, "Concerning the smallest integer m! divisible by a given integer n". ]
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.
Eric Weisstein's World of Mathematics, Smarandache Function.
Index entries for sequences related to factorial numbers.

Cf. A102469, A000522, A002034, A061354, A096058, A007917.

Do[l = FactorInteger[Numerator[Sum[1/k!, {k, 0, n}]]]; If[Length[l] == Plus @@ Last /@ l, Print[Max[First /@ l]], Print[l]], {n, 1, 30}] (* Ryan Propper, Jul 31 2005 *)
nmax = 30; Clear[a]; Do[f = FactorInteger[ Numerator[ Sum[1/k!, {k, 0, n}] ] ]; a[n] = If[Length[f] == Total[f[[All, 2]] ], Max[f[[All, 1]] ], f[[-1, 1]] ], {n, 0, nmax}]; a[3] = 4; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Sep 16 2015, adapted from Ryan Propper's script *)

a(n) = {j = 1; s = numerator(sum(k=0, n, 1/k!)); while (j! % s, j++); j;} \\ Michel Marcus, Sep 16 2015

a(n) = A002034(A061354(n)).

More terms from Ryan Propper, Jul 31 2005

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

Original entry on oeis.org

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

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

A323274 a(n) = ceiling(1/(e - 1/0! - 1/1! - 1/2! - ... - 1/n!)).

Original entry on oeis.org

1, 2, 5, 20, 101, 620, 4420, 35894, 326946, 3301574, 36613057, 442369756, 5784470466, 81391912093, 1226260443926, 19696254286261, 335987466998509, 6066332690596289, 115577941857034741, 2317310520602816401, 48773396185794559169, 1075223007090667361164

Offset: 0

Clark Kimberling, Jan 11 2019

nonn

a(n) = least k such that 1/k > e - (n-th partial sum of the Maclaurin series for e). Let b(n) = a(n)/a(n+1). Conjectures: if n > 3, then n+1 < b(n) < n+2 and 0 < b(n+1)-b(n) < 1.

			Approximates for the first 5 numbers e - (1/0!+1/1!+1/2!+...+1/n!) are 1.71828, 0.718282, 0.218282, 0.0516152, 0.0099485, with approximate reciprocals 0.581977, 1.39221, 4.58123, 19.3742, 100.518.

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A000142, A001113, A061354.

s[n_] := E - Sum[1/k!, {k, 0, n}]
Table[Ceiling[1/s[n]], {n, 0, 30}]

A354138 a(n) is the numerator of Sum_{k=0..n} (-1)^k / (2*k)!.

Original entry on oeis.org

1, 1, 13, 389, 4357, 1960649, 258805669, 47102631757, 11304631621681, 691843455246877, 1314502564969066301, 607300185015708631061, 335229702128671164345673, 217899306383636256824687449, 32946375125205802031892742289, 848027998784883070051677094421

Offset: 0

Ilya Gutkovskiy, May 24 2022

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

Cf. A010050, A049470, A053557, A061354, A103816, A120265, A143382, A354211, A354332, A354334, A354378 (denominators).

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

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

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

A109621 Numbers n such that the numerator of Sum_{k=0..n} 1/k!, in reduced form, is prime.

Original entry on oeis.org

1, 2, 5, 9, 24, 32, 321, 343, 352, 511, 685, 807, 966, 1079, 1274, 1381, 2016, 3226, 8130

Offset: 1

Ryan Propper, Aug 01 2005

Terms through 807 correspond to certified primes.

If it exists, a(20) > 14304. - J.W.L. (Jan) Eerland, Sep 13 2022

			Sum_{k=0..9} 1/k! = 98641/36288 and 98641 is prime, so 9 is in the sequence.

Cf. A061354.

s = 0; Do[s += 1/n!; k = Numerator[s]; If[PrimeQ[k], Print[n]], {n, 0, 3300}]
Flatten[Position[Accumulate[1/Range[0,3230]!],?(PrimeQ[ Numerator[ #]]&)]] -1 (* _Harvey P. Dale, Sep 25 2019 *)
n=0;Monitor[Parallelize[While[True,If[PrimeQ[Numerator[Sum[1/Factorial[k],{k,0,n}]]],Print[n]];n++];n],n] (* J.W.L. (Jan) Eerland, Sep 13 2022 *)

a(19) from J.W.L. (Jan) Eerland, Sep 13 2022

cp's OEIS Frontend

A330030 Least k such that Sum_{i=0..n} k^n / i! is a positive integer.

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A354302 a(n) is the numerator of Sum_{k=0..n} 1 / (k!)^2.

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A354304 a(n) is the numerator of Sum_{k=0..n} (-1)^k / (k!)^2.

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A102468 a(n)! is the smallest factorial divisible by the numerator of Sum_{k=0...n} 1/k!, with a(0) = 1.

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

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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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A323274 a(n) = ceiling(1/(e - 1/0! - 1/1! - 1/2! - ... - 1/n!)).

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