A061355 -id:A061355 - cp's OEIS frontend

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

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}]

A354299 a(n) is the denominator of Sum_{k=1..n} (-1)^(k+1) / (2*k-1)!!.

Original entry on oeis.org

1, 3, 15, 105, 189, 10395, 135135, 2027025, 34459425, 130945815, 13749310575, 316234143225, 7905853580625, 12556355686875, 1238056670725875, 776918153694375, 6332659870762850625, 7642865361265509375, 8200794532637891559375, 63966197354575554163125, 13113070457687988603440625

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, A053556, A061355, A064647, A113013, A143383, A289488, A306858, A354298 (numerators).

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

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

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

A354401 a(n) is the denominator of Sum_{k=1..n} 1 / (k*k!).

Original entry on oeis.org

1, 4, 36, 288, 7200, 10800, 66150, 33868800, 914457600, 4572288000, 553246848000, 737662464000, 41554985472000, 54540918432000, 19634730635520000, 5026491042693120000, 1452655911338311680000, 39221709606134415360000, 14159037167814523944960000, 141590371678145239449600000

Offset: 1

Ilya Gutkovskiy, May 25 2022

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

Cf. A001563, A053556, A061355, A229837, A353545 (numerators), A354404.

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

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

from math import factorial
from fractions import Fraction
def A354401(n): return sum(Fraction(1, k*factorial(k)) for k in range(1,n+1)).denominator # Chai Wah Wu, May 27 2022

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

A354404 a(n) is the denominator of Sum_{k=1..n} (-1)^(k+1) / (k*k!).

Original entry on oeis.org

1, 4, 36, 288, 7200, 10800, 264600, 33868800, 914457600, 4572288000, 553246848000, 2212987392000, 373994869248000, 327245510592000, 19634730635520000, 5026491042693120000, 1452655911338311680000, 39221709606134415360000, 14159037167814523944960000, 141590371678145239449600000

Offset: 1

Ilya Gutkovskiy, May 25 2022

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

Cf. A001563, A053556, A061355, A239069, A354401, A354402 (numerators).

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

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

from math import factorial
from fractions import Fraction
def A354404(n): return sum(Fraction(1 if k & 1 else -1, k*factorial(k)) for k in range(1,n+1)).denominator # Chai Wah Wu, May 27 2022

Denominators 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}]

A102581 Numbers n such that denominator of Sum_{k=0 to 2n+1} 1/k! is (2n+1)!/2.

Original entry on oeis.org

1, 2, 6, 10, 30, 32, 42, 46, 56, 62, 70, 80, 82, 96, 120, 122, 136, 150, 160, 162, 170, 172, 176, 186, 192, 196, 200, 210, 222, 230, 236, 252, 262, 266, 276, 290, 292, 300, 302, 306, 312, 326, 356, 366, 380, 382, 400, 416, 422, 426, 452, 460, 486, 490, 496, 500

Offset: 1

Jonathan Sondow, Jan 21 2005

nonn

The denominator of Sum_{k=0 to m} 1/k! is m!/d, where d = A093101(m). If m > 1 is odd, say m = 2n+1, then d is even. n is a member when d = 2. If m > 3 and m = 3 (mod 4), so that n > 1 is odd, then d is divisible by 4. So except for 1 the members are even.

			1/0! + 1/1! + 1/2! + 1/3! = 8/3 and 3 = (2*1+1)!/2, so 1 is 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

n is a member <=> A093101(2n+1) = 2 <=> A061355(2n+1) = (2n+1)!/2 <=> n = 1 or n/2 is a member of A102582.

fQ[n_] := (Denominator[ Sum[1/k!, {k, 0, 2n + 1}]] == (2n + 1)!/2); Select[ Range[ 500], fQ[ # ] &] (* Robert G. Wilson v, Jan 24 2005 *)

a(n) = 2*A102582(n-1) for n > 1.

More terms from Robert G. Wilson v, Jan 24 2005

A102582 Numbers n such that denominator of Sum_{k=0..4n+1} 1/k! is (4n+1)!/2.

Original entry on oeis.org

1, 3, 5, 15, 16, 21, 23, 28, 31, 35, 40, 41, 48, 60, 61, 68, 75, 80, 81, 85, 86, 88, 93, 96, 98, 100, 105, 111, 115, 118, 126, 131, 133, 138, 145, 146, 150, 151, 153, 156, 163, 178, 183, 190, 191, 200, 208, 211, 213, 226, 230, 243, 245, 248, 250, 256, 260, 261, 265

Offset: 1

Jonathan Sondow, Jan 21 2005

nonn

The denominator of Sum_{k=0 to m} 1/k! is m!/d, where d = A093101(m). If m = 4n+1 > 1, then d is even. n is a member when d = 2.

			1/0! + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! = 163/60 and 60 = 5!/2 = (4*1+1)!/2, so 1 is 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

n is a member <=> 2n is a member of A102581 <=> A093101(4n+1) = 2 <=> A061355(4n+1) = (4n+1)!/2.

fQ[n_] := (Denominator[ Sum[1/k!, {k, 0, 4n + 1}]] == (4n + 1)!/2); Select[ Range[0, 274], fQ[ # ] &] (* Robert G. Wilson v, Jan 24 2005 *)

a(n) = A102581(n+1)/2.

More terms from Robert G. Wilson v, Jan 24 2005

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

Original entry on oeis.org

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)).

A354303 a(n) is the denominator of Sum_{k=0..n} 1 / (k!)^2.

Original entry on oeis.org

1, 1, 4, 18, 576, 2400, 518400, 12700800, 541900800, 65840947200, 13168189440000, 88519495680000, 229442532802560000, 19387894021816320000, 2533351485517332480000, 855006126362099712000000, 437763136697395052544000000, 1621968544942912438272000000

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, A053556, A061355, A070910, A143383, A354302 (numerators), A354305.

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

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

A354305 a(n) is the denominator of Sum_{k=0..n} (-1)^k / (k!)^2.

Original entry on oeis.org

1, 1, 4, 9, 192, 1800, 103680, 529200, 232243200, 8230118400, 1463132160000, 39833773056000, 20858412072960000, 1615657835151360000, 584619573580922880000, 1908495817772544000000, 29184209113159670169600000, 3953548328298349068288000000, 185476873609942457647104000000

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, A053556, A061355, A073701, A091681, A143383, A354303, A354304 (numerators).

Table[Sum[(-1)^k/(k!)^2, {k, 0, n}], {n, 0, 18}] // Denominator
nmax = 18; CoefficientList[Series[BesselJ[0, 2 Sqrt[x]]/(1 - x), {x, 0, nmax}], x] // Denominator
Accumulate[Table[(-1)^k/(k!)^2,{k,0,20}]]//Denominator (* Harvey P. Dale, Apr 25 2023 *)

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

cp's OEIS Frontend

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

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A354299 a(n) is the denominator of Sum_{k=1..n} (-1)^(k+1) / (2*k-1)!!.

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A354401 a(n) is the denominator of Sum_{k=1..n} 1 / (k*k!).

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A354404 a(n) is the denominator of Sum_{k=1..n} (-1)^(k+1) / (k*k!).

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

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A102581 Numbers n such that denominator of Sum_{k=0 to 2n+1} 1/k! is (2n+1)!/2.

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A102582 Numbers n such that denominator of Sum_{k=0..4n+1} 1/k! is (4n+1)!/2.

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

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