A120265 -id:A120265 - cp's OEIS frontend

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

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

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).

A129924 Primes p such that p divides both A061354(p-3) and A061354(p-1).

Original entry on oeis.org

5, 13, 37, 463

Offset: 1

Alexander Adamchuk, Jun 06 2007

Conjecture: a(n) = A064384(n+1).
Also primes p such that p divides A120265(p-2), where A120265(n) = A061354(n) - A061355(n) = Numerator of Sum[1/k!,{k,1,n}].
The conjecture is true. It is the case n = p-3 of the relation GCD(A061354(n), A061354(n+2)) = A124779(n), which follows from the Comments in A064384 and A124779. For a proof, see the link "The Taylor series for e ...". - Jonathan Sondow, Jun 12 2007
Michael Mossinghoff has calculated that 5, 13, 37, 463 are the only terms up to 150 million. Heuristics suggest the sequence is infinite but very sparse. - Jonathan Sondow, Jun 12 2007

J. Sondow, The Taylor series for e and the primes 2, 5, 13, 37, 463, ...: a surprising connection
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. A061354 = Numerator of Sum_{k=0..n} 1/k!. Cf. A064384, A124779.

Cf. A120265 = Numerator of Sum[1/k!, {k, 1, n}]. Cf. A061355.

g=1; Do[ g=g+1/n!; f=Numerator[g]; If[ PrimeQ[n+3] && IntegerQ[f/(n+3)], Print[n+3]], {n,1,1000}]

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.

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).

cp's OEIS Frontend

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

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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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A129924 Primes p such that p divides both A061354(p-3) and A061354(p-1).

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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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