A061354 -id:A061354 - cp's OEIS frontend

A123900 a(n) = (n+3)!/(d(n)d(n+1)d(n+2)) where d(n) = cancellation factor in reducing Sum_{k=0...n} 1/k! to lowest terms.

6, 12, 60, 180, 2520, 1008, 18144, 18144, 3991680, 5987520, 155675520, 1089728640, 26153487360, 523069747200, 17784371404800, 12312257126400, 935731541606400, 4678657708032, 12772735542927360, 140500090972200960

Offset: 0

Jonathan Sondow, Oct 18 2006

			a(2) = 60 because (2+3)!/(d(2)*d(3)*d(4)) = 5!/(GCD(2,5)*GCD(6,16)*GCD(24,65)) = 120/2 = 60.

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.

Cf. A000522, A061354, A093101, A123899, A123901.

(A[n_] := If[n==0,1,n*A[n-1]+1]; d[n_] := GCD[A[n],n! ]; Table[(n+3)!/(d[n]*d[n+1]*d[n+2]), {n,0,21}])

a(n) = (n+3)!/(A093101(n)*A093101(n+1)*A093101(n+2)) where A093101(n) = gcd(n!,1+n+n(n-1)+...+n!).

A069880 Number of terms in the simple continued fraction for Sum_{k=1..n} 1/k!.

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 9, 13, 14, 18, 19, 20, 24, 24, 23, 29, 33, 36, 31, 38, 41, 42, 46, 50, 53, 58, 56, 57, 70, 73, 77, 69, 76, 76, 78, 77, 80, 85, 89, 101, 101, 105, 106, 104, 106, 112, 115, 124, 113, 126, 124, 124, 130, 144, 144, 148, 140, 149, 141, 151, 157, 158, 172

Offset: 1

Benoit Cloitre, May 04 2002

			For n=4, Sum_{k=1..n} 1/k! = 1/1! + 1/2! + 1/3! + 1/4! = 1/1 + 1/2 + 1/6 + 1/24 = 41/24 = 1 + 1/(1 + 1/(2 + 1/(2 + 1/3))) = CF[1;1,2,2,1], so a(4) = 5.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Amiram Eldar, Plot of a(n)/(n * log(log(n))) for n = 1..10000.

Cf. A055573, A061354, A061355, A070267.

lcf[f_] := Length[ContinuedFraction[f]]; lcf /@ Accumulate[Table[1/k!, {k, 1, 100}]] (* Amiram Eldar, Apr 30 2022 *)

Does lim_{n->infinity} a(n)/(n * log(log(n))) = C = 2.XXX...?

A354211 a(n) is the numerator of Sum_{k=0..n} 1 / (2*k+1)!.

Original entry on oeis.org

1, 7, 47, 5923, 426457, 15636757, 7318002277, 1536780478171, 603180793741, 142957467201379447, 60042136224579367741, 10127106976545720025649, 18228792557782296046168201, 12796612375563171824410077103, 3463616416319098507140327535879, 1380498543075754976417359117871773

Offset: 0

Ilya Gutkovskiy, May 24 2022

			1, 7/6, 47/40, 5923/5040, 426457/362880, 15636757/13305600, 7318002277/6227020800, ...

Cf. A009445, A053557, A061354, A073742, A103816, A120265, A143382, A289381, A354331 (denominators), A354332, A354334.

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

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

from fractions import Fraction
from math import factorial
def A354211(n): return sum(Fraction(1,factorial(2*k+1)) for k in range(n+1)).numerator # Chai Wah Wu, May 24 2022

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

A354332 a(n) is the numerator of Sum_{k=0..n} (-1)^k / (2*k+1)!.

Original entry on oeis.org

1, 5, 101, 4241, 305353, 33588829, 209594293, 1100370038249, 23023126954133, 102360822438075317, 42991545423991633141, 4350744396907953273869, 13052233190723859821607001, 9162667699888149594768114701, 7440086172309177470951709137213, 364172638960396581472899447242531

Offset: 0

Ilya Gutkovskiy, May 24 2022

			1, 5/6, 101/120, 4241/5040, 305353/362880, 33588829/39916800, 209594293/249080832, ...

Cf. A009445, A049469, A053557, A061354, A103816, A120265, A143382, A354211, A354298, A354333 (denominators), A354334.

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

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

from fractions import Fraction
from math import factorial
def A354332(n): return sum(Fraction(-1 if k % 2 else 1,factorial(2*k+1)) for k in range(n+1)).numerator # Chai Wah Wu, May 24 2022

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

A354334 a(n) is the numerator of Sum_{k=0..n} 1 / (2*k)!.

Original entry on oeis.org

1, 3, 37, 1111, 6913, 799933, 739138093, 44841044309, 32285551902481, 9879378882159187, 1251387991740163687, 1734423756551866870183, 136771701945232930334431, 23048564587067030852654113, 42769754577382930342215977687, 409306551305554643375006906464591

Offset: 0

Ilya Gutkovskiy, May 24 2022

			1, 3/2, 37/24, 1111/720, 6913/4480, 799933/518400, 739138093/479001600, ...

Cf. A010050, A053557, A061354, A073743, A103816, A120265, A143382, A354211, A354332, A354335 (denominators).

Table[Sum[1/(2 k)!, {k, 0, n}], {n, 0, 15}] // Numerator
nmax = 15; CoefficientList[Series[Cosh[Sqrt[x]]/(1 - x), {x, 0, nmax}], x] // Numerator
Accumulate[1/(2*Range[0,20])!]//Numerator (* Harvey P. Dale, Sep 05 2024 *)

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

from fractions import Fraction
from math import factorial
def A354334(n): return sum(Fraction(1,factorial(2*k)) for k in range(n+1)).numerator # Chai Wah Wu, May 24 2022

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

A354298 a(n) is the numerator of Sum_{k=1..n} (-1)^(k+1) / (2*k-1)!!.

Original entry on oeis.org

1, 2, 11, 76, 137, 7534, 97943, 1469144, 24975449, 94906706, 9965204131, 229199695012, 5729992375301, 9100576125478, 897316805972131, 563093542209232, 4589775462547450033, 5539384178936577626, 5943759223998947792699, 46361321947191792783052, 9504070999174317520525661

Offset: 1

Ilya Gutkovskiy, May 23 2022

			1, 2/3, 11/15, 76/105, 137/189, 7534/10395, 97943/135135, 1469144/2027025, 24975449/34459425, ...

Robert Israel, Table of n, a(n) for n = 1..403

Cf. A001147, A053557, A061354, A064646, A103816, A113012, A120265, A143382, A289381, A306858, A354299 (denominators).

S:= 0: R:= NULL:
for n from 1 to 100 do
  S:= S + (-1)^(n+1)/doublefactorial(2*n-1);
  R:= R, numer(S);
od:
R; # Robert Israel, Jan 10 2024

Table[Sum[(-1)^(k + 1)/(2 k - 1)!!, {k, 1, n}], {n, 1, 21}] // Numerator
nmax = 21; CoefficientList[Series[Sqrt[Pi x Exp[-x]/2] Erfi[Sqrt[x/2]]/(1 - x), {x, 0, nmax}], x] // Numerator // Rest
Table[1/(1 + ContinuedFractionK[2 k - 1, 2 k, {k, 1, n - 1}]), {n, 1, 21}] // Numerator

Numerators of coefficients in expansion of sqrt(Pi*x*exp(-x)/2) * erfi(sqrt(x/2)) / (1 - x).

A353545 a(n) is the numerator of Sum_{k=1..n} 1 / (k*k!).

Original entry on oeis.org

1, 5, 47, 379, 9487, 14233, 87179, 44635753, 1205165611, 6025828181, 729125211161, 972166948343, 54765404757169, 71879593743829, 25876653747779441, 6624423359431551911, 1914458350875718742519, 51690375473644406388353, 18660225545985630712321553, 186602255459856307126125437

Offset: 1

Ilya Gutkovskiy, May 25 2022

			1, 5/4, 47/36, 379/288, 9487/7200, 14233/10800, 87179/66150, ...

Cf. A001563, A053557, A061354, A103816, A120265, A229837, A354401 (denominators), A354402.

Table[Sum[1/(k k!), {k, 1, n}], {n, 1, 20}] // Numerator
nmax = 20; Assuming[x > 0, CoefficientList[Series[(ExpIntegralEi[x] - Log[x] - EulerGamma)/(1 - x), {x, 0, nmax}], x]] // Numerator // Rest

a(n) = numerator(sum(k=1, n, 1/(k*k!))); \\ Michel Marcus, May 26 2022

from math import factorial
from fractions import Fraction
def A353545(n): return sum(Fraction(1, k*factorial(k)) for k in range(1,n+1)).numerator # Chai Wah Wu, May 27 2022

Numerators of coefficients in expansion of (Ei(x) - log(x) - gamma) / (1 - x), x > 0.

A354402 a(n) is the numerator of Sum_{k=1..n} (-1)^(k+1) / (k*k!).

Original entry on oeis.org

1, 3, 29, 229, 5737, 8603, 210781, 26979863, 728456581, 3642282779, 440716217519, 1762864869691, 297924162982399, 260683642609331, 15641018556560861, 4004100750479565401, 1157185116888594641129, 31243998155992054970143, 11279083334313131850347743, 112790833343131318500567523

Offset: 1

Ilya Gutkovskiy, May 25 2022

			1, 3/4, 29/36, 229/288, 5737/7200, 8603/10800, 210781/264600, ...

Cf. A001563, A053557, A061354, A103816, A120265, A239069, A353545, A354404 (denominators).

Table[Sum[(-1)^(k + 1)/(k k!), {k, 1, n}], {n, 1, 20}] // Numerator
nmax = 20; Assuming[x > 0, CoefficientList[Series[(EulerGamma + Log[x] - ExpIntegralEi[-x])/(1 - x), {x, 0, nmax}], x]] // Numerator // Rest

a(n) = numerator(sum(k=1, n, (-1)^(k+1)/(k*k!))); \\ Michel Marcus, May 26 2022

from math import factorial
from fractions import Fraction
def A354402(n): return sum(Fraction(1 if k & 1 else -1, k*factorial(k)) for k in range(1,n+1)).numerator # Chai Wah Wu, May 27 2022

Numerators of coefficients in expansion of (gamma + log(x) - Ei(-x)) / (1 - x), x > 0.

A070267 Maximum element in the simple continued fraction expansion of e(n) = 1+1/2!+1/3!+...+1/n!.

Original entry on oeis.org

1, 2, 2, 3, 8, 5, 4, 14, 6, 29, 10, 16, 20, 18, 42, 59, 13, 14, 59, 35, 31, 184, 24, 65, 42, 64, 401, 71, 26, 24, 36, 31, 52, 187, 28, 41, 128, 177, 3041, 249, 315, 162, 118, 36, 101, 135, 86, 70, 194, 104, 274, 62, 2515, 305, 68, 59, 49, 88, 359, 280, 100, 702, 52

Offset: 1

Benoit Cloitre, May 09 2002

			The simple continued fraction expansion of e(10) is [1, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 11, 1, 1, 29, 1, 1, 2], hence a(10) = 29.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A061354, A061355, A069880, A070266.

Table[ Max[ ContinuedFraction[ Sum[1/i!, {i, 1, n}]]], {n, 1, 65}]

A102469 Largest prime factor of numerator of Sum_{k=0...n} 1/k!, with a(0) = 1.

Original entry on oeis.org

1, 2, 5, 2, 13, 163, 103, 137, 863, 98641, 10687, 31469, 1540901, 522787, 5441, 226871807, 13619, 1276861, 414026539, 2124467, 12670743557, 838025081381, 44659157, 323895443, 337310723185584470837549, 54352957, 11301647941785046703319941, 102505951982728548829

Offset: 0

Jonathan Sondow, Jan 09 2005

nonn

It appears that a(n) = A102468(n) (Smarandache number of the same numerator) except when n = 3. The largest prime factor of the corresponding denominator is A007917(n) for n > 1. Omitting the 0th term in the sum, it appears that the largest prime factor and the Kempner number A002034, of the numerator of Sum_{k=1...n} 1/k! are both equal to A096058(n).

			Sum_{k=0...3} 1/k! = 8/3 and 2 is the largest prime factor 8, so a(3) = 2.

Daniel Suteu, Table of n, a(n) for n = 0..74
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, Greatest Prime Factor
Index entries for sequences related to factorial numbers.

Cf. A102468, A096058, A006530, A061354, A000522, A007917.

FactorInteger[#][[-1,1]]&/@Numerator[Accumulate[1/Range[0,30]!]] (* Harvey P. Dale, Nov 14 2012 *)

a(n) = if(n==0, return(1)); vecmax(factor(numerator(sum(k=0, n, 1/k!)))[,1]); \\ Daniel Suteu, Jun 09 2022

a(n) = A006530(A061354(n)).

cp's OEIS Frontend

A123900 a(n) = (n+3)!/(d(n)*d(n+1)*d(n+2)) where d(n) = cancellation factor in reducing Sum_{k=0...n} 1/k! to lowest terms.

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A069880 Number of terms in the simple continued fraction for Sum_{k=1..n} 1/k!.

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A354211 a(n) is the numerator of Sum_{k=0..n} 1 / (2*k+1)!.

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A354332 a(n) is the numerator of Sum_{k=0..n} (-1)^k / (2*k+1)!.

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A354334 a(n) is the numerator of Sum_{k=0..n} 1 / (2*k)!.

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A354298 a(n) is the numerator of Sum_{k=1..n} (-1)^(k+1) / (2*k-1)!!.

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A353545 a(n) is the numerator of Sum_{k=1..n} 1 / (k*k!).

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A354402 a(n) is the numerator of Sum_{k=1..n} (-1)^(k+1) / (k*k!).

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A123900 a(n) = (n+3)!/(d(n)d(n+1)d(n+2)) where d(n) = cancellation factor in reducing Sum_{k=0...n} 1/k! to lowest terms.