A282529 -id:A282529 - cp's OEIS frontend

A247551 Decimal expansion of Product_{k>=2} 1/(1-1/k!).

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

Offset: 1

Vaclav Kotesovec, Sep 19 2014

			2.5294774720791526481801161542539542411787023948459973375849349825...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 269.
A. Knopfmacher, A. M. Odlyzko, B. Pittel, L. B. Richmond, D. Stark, G. Szekeres, and N. C. Wormald, The Asymptotic Number of Set Partitions with Unequal Block Sizes, The Electronic Journal of Combinatorics, Volume 6 (1999), Research Paper #R2.

Cf. A005651, A035341, A072605, A140585, A183239, A209668, A261600, A282529, A292713, A320566, A327827, A335643.

evalf(1/product(1-1/k!,k=2..infinity),100); # Vaclav Kotesovec, Sep 19 2014

digits = 105;
RealDigits[NProduct[1/(1-1/k!), {k, 2, Infinity}, WorkingPrecision -> digits+10, NProductFactors -> digits], 10, digits][[1]] (* Jean-François Alcover, Nov 02 2020 *)

default(realprecision,150); 1/prodinf(k=2,1 - 1/k!) \\ Vaclav Kotesovec, Sep 21 2014

Product_{k>=2} 1/(1-1/k!).
Equals lim_{n -> oo} A005651(n) / n!.
Equals 1/A282529. - Amiram Eldar, Sep 15 2023

A238695 Decimal expansion of Product_{k>=0} (1+1/k!).

Original entry on oeis.org

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

Offset: 1

Frederick Reckless, Mar 03 2014

Conjectured to be irrational, transcendental and normal, none have been shown. Product is sometimes taken from n=1, leading to half the stated value.

			7.3643082723672572563727725096310530956542568360689...

Lucian Craciun, Table of n, a(n) for n = 1..10000
StackExchange, Is Product_{n>=0} (1+1/n!) = e^2?, Apr 06 2013
StackExchange, How to compute Product_{n>=1} (1+1/n!)?, Aug 05 2013
Wolfram Alpha, Product_{n>=0} (1+1/n!)

Cf. A072334, A217757, A282529.

Cf. A387175 (continued fraction).

evalf(product(1+1/n!, n = 0..infinity), 100) # Lucian Craciun, Feb 17 2017

RealDigits[Product[1+1/n!, {n, 0, 75}], 10, 106][[1]] (* Robert G. Wilson v, Mar 19 2014 *)

prodinf(n=1,1+1/gamma(n)) \\ Charles R Greathouse IV, Nov 13 2014

Added more digits from b-file, so as to cover exactly three full rows of text. - Lucian Craciun, Feb 22 2017

A369994 Decimal expansion of Product_{k>=2} (1 - 1/k!!).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Mar 30 2024

			0.26294413828698825249397612947579574217...

Cf. A006882, A282529, A370413.

A370424 Decimal expansion of Product_{k>=2} (1 - 1/(2*k-1)!!).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Mar 30 2024

			0.61558002800838914113852450027455396226...

Cf. A001147, A282529, A370465.

A342574 Remainder of e when the Dirichlet series for e is converted to an infinite product (negated).

Original entry on oeis.org

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

Offset: 0

Michael P. May, Mar 27 2021

Remainder term when the Dirichlet series representation of e is converted to an infinite product via the Euler Product Formula method. The remainder can be calculated via the infinite series found at the top of page 2 of "A Non-Sieving Application ..." (see links) or by the much simpler method A001113*A282529-2.

			-0.925358356236040633370884166370763828049565...

Michael P. May, A Non-Sieving Application of the Euler Zeta Function, arXiv:1510.01549 [math.NT], 2015.

Cf. A001113 (e), A282529.

Equals A001113*A282529-2.

More digits from Alois P. Heinz, Mar 31 2021

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A247551 Decimal expansion of Product_{k>=2} 1/(1-1/k!).

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A238695 Decimal expansion of Product_{k>=0} (1+1/k!).

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A369994 Decimal expansion of Product_{k>=2} (1 - 1/k!!).

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A370424 Decimal expansion of Product_{k>=2} (1 - 1/(2*k-1)!!).

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A342574 Remainder of e when the Dirichlet series for e is converted to an infinite product (negated).

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