A288091 -id:A288091 - cp's OEIS frontend

A007662 Quadruple factorial numbers n!!!!: a(n) = n*a(n-4).

1, 1, 2, 3, 4, 5, 12, 21, 32, 45, 120, 231, 384, 585, 1680, 3465, 6144, 9945, 30240, 65835, 122880, 208845, 665280, 1514205, 2949120, 5221125, 17297280, 40883535, 82575360, 151412625, 518918400, 1267389585, 2642411520, 4996616625

Offset: 0

N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. Spanier and K. B. Oldham, An Atlas of Functions, Hemisphere, NY, 1987, p. 23.

Klaus Brockhaus, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Multifactorial.

Cf. A047053, A007696, A001813, A008545, A000142, A006882, A007661, A288091.

I:=[ 1, 1, 2, 3 ]; [ n le 4 select I[n] else (n-1)*Self(n-4): n in [1..36] ]; // Klaus Brockhaus, Jun 23 2011

A007662:=func< n | n eq 0 select 1 else &*[ k: k in [1..n] | k mod 4 eq n mod 4 ] >; [ A007662(n): n in [0..35] ]; // Klaus Brockhaus, Jun 23 2011

NFactorialM[n_Integer, m_Integer] := Block[{k = n, p = Max[1, n]}, While[k > m, k -= m; p *= k]; p]; Table[ NFactorialM[n, 4], {n, 0, 34}] (* Robert G. Wilson v *)
With[{k = 4}, Table[With[{q = Quotient[n + k - 1, k]}, k^q q! Binomial[n/k, q]], {n, 0, 34}]] (* Jan Mangaldan, Mar 21 2013 *)

a(n)=if(n<6,max(n,1),n*a(n-4)) \\ Charles R Greathouse IV, Jun 23 2011

a(n) ~ c * n^(n/4+1/2)/exp(n/4), where c = sqrt(Pi/2) if n=4*k, c = 2*sqrt(Pi)/Gamma(1/4) if n=4*k+1, c = sqrt(2) if n=4*k+2, c = sqrt(Pi)/Gamma(3/4) if n=4*k+3. - Vaclav Kotesovec, Jul 29 2013

Sum_{n>=0} 1/a(n) = A288091. - Amiram Eldar, Nov 10 2020

A143280 Decimal expansion of m(2) = Sum_{n>=0} 1/n!!.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Aug 04 2008

Also decimal expansion of Sum_{n>=1} n!!/n!. - Michel Lagneau, Dec 24 2011

Apart from the first digit, the same as A227569. - Robert G. Wilson v, Apr 09 2014

			3.05940740534257614453947549923327861297725472633534020929971877980544281968...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Michael Penn, Finding the closed form for a double factorial sum, YouTube video, 2022.
Eric Weisstein's World of Mathematics, Double Factorial
Eric Weisstein's World of Mathematics, Reciprocal Multifactorial Constant

Cf. A227569.

Cf. A006882 (n!!), this sequence (m(2)), A288055 (m(3)), A288091 (m(4)), A288092 (m(5)), A288093 (m(6)), A288094 (m(7)), A288095 (m(8)), A288096 (m(9)).

SetDefaultRealField(RealField(100)); R:= RealField(); Exp(1/2)*(1 + Sqrt(Pi(R)/2)*Erf(1/Sqrt(2) )); // G. C. Greubel, Mar 27 2019

RealDigits[ Sqrt[E] + Sqrt[E*Pi/2]*Erf[1/Sqrt[2]], 10, 105][[1]] (* or *)
RealDigits[ Sum[1/n!!, {n, 0, 125}], 10, 105][[1]] (* Robert G. Wilson v, Apr 09 2014 *)
RealDigits[Total[1/Range[0,200]!!],10,120][[1]] (* Harvey P. Dale, Apr 10 2022 *)

default(realprecision, 100); exp(1/2)*(1 + sqrt(Pi/2)*(1-erfc(1/sqrt(2) ))) \\ G. C. Greubel, Mar 27 2019

numerical_approx(exp(1/2)*(1 + sqrt(pi/2)*erf(1/sqrt(2))), digits=100) # G. C. Greubel, Mar 27 2019

Equals sqrt(e) + sqrt((e*Pi)/2)*erf(1/sqrt(2)).

A288055 Decimal expansion of m(3) = Sum_{n>=0} 1/n!!!, the 3rd reciprocal multifactorial constant.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 05 2017

			3.298913538088419003401217808261469769077803695683207090885045129...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Reciprocal Multifactorial Constant

Cf. A007661 (n!!!), A143280 (m(2)), this sequence (m(3)), A288091 (m(4)), A288092 (m(5)), A288093 (m(6)), A288094 (m(7)), A288095 (m(8)), A288096 (m(9)).

SetDefaultRealField(RealField(100)); (Exp(1/3)/3)*(3 + 3^(1/3)*Gamma(1/3, 1/3) + 3^(2/3)*Gamma(2/3, 1/3)); // G. C. Greubel, Mar 27 2019

p = Pochhammer;
m[3] = 5/2 + Sum[1/((-3)^k*((2+3*k)*p[1/3 -k, k])) + 1/((-3)^k*((1+3*k) *p[2/3 -k, k])) +(-1)^(1-k)/(3^k*(k*p[1-k, -1 +k])), {k, 1, Infinity}]
(* or *)
m[3] = (Exp[1/3]/3) (3 + 3^(1/3) (Gamma[1/3] - Gamma[1/3, 1/3]) + 3^(2/3) (Gamma[2/3] - Gamma[2/3, 1/3]));
RealDigits[m[3], 10, 102][[1]]

default(realprecision, 100); (exp(1/3)/3)*(3 + 3^(1/3)*(gamma(1/3) - incgam(1/3, 1/3)) + 3^(2/3)*(gamma(2/3) - incgam(2/3, 1/3))) \\ G. C. Greubel, Mar 27 2019

numerical_approx((exp(1/3)/3)*(3 + 3^(1/3)*(gamma(1/3) - gamma_inc(1/3, 1/3)) + 3^(2/3)*(gamma(2/3) - gamma_inc(2/3, 1/3))), digits=100) # G. C. Greubel, Mar 27 2019

m(k) = (1/k)*exp(1/k)*(k + Sum_{j=1..k-1} k^(j/k)*(gamma(j/k) - gamma(j/k, 1/k))) where gamma(x) is the Euler gamma function and gamma(a,x) the incomplete gamma function.

A288092 Decimal expansion of m(5) = Sum_{n>=0} 1/n!5, the 5th reciprocal multifactorial constant.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 05 2017

			3.640224467733809734176937236963569060632409516968842599452955763...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Reciprocal Multifactorial Constant

Cf. A085157 (n!5), A143280 (m(2)), A288055 (m(3)), A288091 (m(4)), this sequence (m(5)), A288093 (m(6)), A288094 (m(7)), A288095 (m(8)), A288096 (m(9)).

SetDefaultRealField(RealField(105)); (1/5)*Exp(1/5)*(5 + (&+[5^(k/5)*Gamma(k/5, 1/5): k in [1..4]])); // G. C. Greubel, Mar 28 2019

m[5] = (1/5)*E^(1/5)*(5 + 5^(1/5)*(Gamma[1/5] - Gamma[1/5, 1/5]) + 5^(2/5)*(Gamma[2/5] - Gamma[2/5, 1/5]) + 5^(3/5)*(Gamma[3/5] - Gamma[3/5, 1/5]) + 5^(4/5)*(Gamma[4/5] - Gamma[4/5, 1/5])); RealDigits[m[5], 10, 102][[1]]

default(realprecision, 105); (1/5)*exp(1/5)*(5 + sum(k=1,4, 5^(k/5)*(gamma(k/5) - incgam(k/5, 1/5)))) \\ G. C. Greubel, Mar 28 2019

numerical_approx((1/5)*exp(1/5)*(5 + sum(5^(k/5)*(gamma(k/5) - gamma_inc(k/5, 1/5)) for k in (1..4))), digits=105) # G. C. Greubel, Mar 28 2019

m(k) = (1/k)*exp(1/k)*(k + Sum_{j=1..k-1} k^(j/k)*(gamma(j/k) - gamma(j/k, 1/k))) where gamma(x) is the Euler gamma function and gamma(a,x) the incomplete gamma function.

A288093 Decimal expansion of m(6) = Sum_{n>=0} 1/n!6, the 6th reciprocal multifactorial constant.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 05 2017

			3.771902396211758435660053589263943632646890281574478369567756485...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Reciprocal Multifactorial Constant

Cf. A085158 (n!6), A143280 (m(2)), A288055 (m(3)), A288091 (m(4)), A288092 (m(5)), this sequence (m(6)), A288094 (m(7)), A288095 (m(8)), A288096 (m(9)).

SetDefaultRealField(RealField(105)); (1/6)*Exp(1/6)*(6 + (&+[6^(k/6)*Gamma(k/6, 1/6): k in [1..5]])); // G. C. Greubel, Mar 28 2019

m[k_] := (1/k) Exp[1/k] (k + Sum[k^(j/k) (Gamma[j/k] - Gamma[j/k, 1/k]), {j, 1, k - 1}]); RealDigits[m[6], 10, 102][[1]]

default(realprecision, 105); (1/6)*exp(1/6)*(6 + sum(k=1,5, 6^(k/6)*(gamma(k/6) - incgam(k/6, 1/6)))) \\ G. C. Greubel, Mar 28 2019

numerical_approx((1/6)*exp(1/6)*(6 + sum(6^(k/6)*(gamma(k/6) - gamma_inc(k/6, 1/6)) for k in (1..5))), digits=105) # G. C. Greubel, Mar 28 2019

m(k) = (1/k)*exp(1/k)*(k + Sum_{j=1..k-1} k^(j/k)*(gamma(j/k) - gamma(j/k, 1/k))) where gamma(x) is the Euler gamma function and gamma(a,x) the incomplete gamma function.

A288094 Decimal expansion of m(7) = Sum_{n>=0} 1/n!7, the 7th reciprocal multifactorial constant.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 05 2017

			3.88695965374084349542856991093670567270530958752016048580439533869...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Reciprocal Multifactorial Constant

Cf. A114799 (n!7), A143280 (m(2)), A288055 (m(3)), A288091 (m(4)), A288092 (m(5)), A288093 (m(6)), this sequence (m(7)), A288095 (m(8)), A288096 (m(9)).

SetDefaultRealField(RealField(105)); (1/7)*Exp(1/7)*(7 + (&+[7^(k/7)*Gamma(k/7, 1/7): k in [1..6]])); // G. C. Greubel, Mar 28 2019

m[k_] := (1/k) Exp[1/k] (k + Sum[k^(j/k) (Gamma[j/k] - Gamma[j/k, 1/k]), {j, 1, k - 1}]); RealDigits[m[7], 10, 104][[1]]
RealDigits[Total[Table[1/Times@@Range[n,1,-7],{n,0,500}]],10,120][[1]] (* Harvey P. Dale, May 21 2023 *)

default(realprecision, 105); (1/7)*exp(1/7)*(7 + sum(k=1,6, 7^(k/7)*(gamma(k/7) - incgam(k/7, 1/7)))) \\ G. C. Greubel, Mar 28 2019

numerical_approx((1/7)*exp(1/7)*(7 + sum(7^(k/7)*(gamma(k/7) - gamma_inc(k/7, 1/7)) for k in (1..6))), digits=105) # G. C. Greubel, Mar 28 2019

m(k) = (1/k)*exp(1/k)*(k + Sum_{j=1..k-1} k^(j/k)*(gamma(j/k) - gamma(j/k, 1/k))) where gamma(x) is the Euler gamma function and gamma(a,x) the incomplete gamma function.

A288095 Decimal expansion of m(8) = Sum_{n>=0} 1/n!8, the 8th reciprocal multifactorial constant.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 05 2017

			3.9892412126901365441336421348019099438359273924576814826209556653...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Reciprocal Multifactorial Constant

Cf. A114800 (n!8), A143280 (m(2)), A288055 (m(3)), A288091 (m(4)), A288092 (m(5)), A288093 (m(6)), A288094 (m(7)), this sequence (m(8)), A288096 (m(9)).

SetDefaultRealField(RealField(107)); (1/8)*Exp(1/8)*(8 + (&+[8^(k/8)*Gamma(k/8, 1/8): k in [1..7]])); // G. C. Greubel, Mar 28 2019

m[k_] := (1/k) Exp[1/k] (k + Sum[k^(j/k) (Gamma[j/k] - Gamma[j/k, 1/k]), {j, 1, k - 1}]); RealDigits[m[8], 10, 103][[1]]

default(realprecision, 105); (1/8)*exp(1/8)*(8 + sum(k=1,7, 8^(k/8)*(gamma(k/8) - incgam(k/8, 1/8)))) \\ G. C. Greubel, Mar 28 2019

numerical_approx((1/8)*exp(1/8)*(8 + sum(8^(k/8)*(gamma(k/8) - gamma_inc(k/8, 1/8)) for k in (1..7))), digits=105) # G. C. Greubel, Mar 28 2019

m(k) = (1/k)*exp(1/k)*(k + Sum_{j=1..k-1} k^(j/k)*(gamma(j/k) - gamma(j/k, 1/k))) where gamma(x) is the Euler gamma function and gamma(a,x) the incomplete gamma function.

A288096 Decimal expansion of m(9) = Sum_{n>=0} 1/n!9, the 9th reciprocal multifactorial constant.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 05 2017

			4.08137552016889854407110514660961069462641007731860758843485175...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Reciprocal Multifactorial Constant

Cf. A114806 (n!9), A143280 (m(2)), A288055 (m(3)), A288091 (m(4)), A288092 (m(5)), A288093 (m(6)), A288094 (m(7)), A288095 (m(8)), this sequence (m(9)).

SetDefaultRealField(RealField(105)); (1/9)*Exp(1/9)*(9 + (&+[9^(k/9)*Gamma(k/9, 1/9): k in [1..8]])); // G. C. Greubel, Mar 28 2019

m[k_] := (1/k) Exp[1/k] (k + Sum[k^(j/k) (Gamma[j/k] - Gamma[j/k, 1/k]), {j, 1, k - 1}]); RealDigits[m[9], 10, 102][[1]]

default(realprecision, 105); (1/9)*exp(1/9)*(9 + sum(k=1,8, 9^(k/9)*(gamma(k/9) - incgam(k/9, 1/9)))) \\ G. C. Greubel, Mar 28 2019

numerical_approx((1/9)*exp(1/9)*(9 + sum(9^(k/9)*(gamma(k/9) - gamma_inc(k/9, 1/9)) for k in (1..8))), digits=105) # G. C. Greubel, Mar 28 2019

m(k) = (1/k)*exp(1/k)*(k + Sum_{j=1..k-1} k^(j/k)*(gamma(j/k) - gamma(j/k, 1/k))) where gamma(x) is the Euler gamma function and gamma(a,x) the incomplete gamma function.

A342033 Decimal expansion of m(10) = Sum_{n>=0} 1/n!10, the 10th reciprocal multifactorial constant.

Original entry on oeis.org

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

Offset: 1

Bhoris Dhanjal, Feb 26 2021

m(k) can be proved to approach a harmonic series (and diverge) as k approaches infinity.

			4.165243765558384590787262...
For n=10, the series is equal to 1+summation from n=1 to 10 (1/n)=9901/2520.

Eric Weisstein's World of Mathematics, Reciprocal Multifactorial Constant

Cf. A143280 (m(2)), A288055 (m(3)), A288091 (m(4)), A288092 (m(5)), A288093 (m(6)), A288094 (m(7)), A288095 (m(8)), A288096 (m(9)).

Multifactorial[n_, k_] := Abs[Apply[Times, Range[-n, -1, k]]]
N[Sum[1/Multifactorial[n, 10], {n, 0, 10000}], 105]
(* or *)
ReciprocalFactorialSumConstant[k_] :=
1/k Exp[1/k] (k + Sum[k^(j/k) Gamma[j/k, 0, 1/k], {j, k - 1}])
N[ReciprocalFactorialSumConstant[10], 105]

m(k) = (1/k)*exp(1/k)*(k + Sum_{j=1..k-1} (k^(j/k)*Gamma(j/k, 1/k))) where Gamma(a,x) the incomplete Gamma function.

cp's OEIS Frontend

A007662 Quadruple factorial numbers n!!!!: a(n) = n*a(n-4).

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A143280 Decimal expansion of m(2) = Sum_{n>=0} 1/n!!.

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A288055 Decimal expansion of m(3) = Sum_{n>=0} 1/n!!!, the 3rd reciprocal multifactorial constant.

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A288092 Decimal expansion of m(5) = Sum_{n>=0} 1/n!5, the 5th reciprocal multifactorial constant.

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A288093 Decimal expansion of m(6) = Sum_{n>=0} 1/n!6, the 6th reciprocal multifactorial constant.

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A288094 Decimal expansion of m(7) = Sum_{n>=0} 1/n!7, the 7th reciprocal multifactorial constant.

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