A238695 -id:A238695 - cp's OEIS frontend

A217757 Product_{i=0..n} (i! + 1).

2, 4, 12, 84, 2100, 254100, 183206100, 923541950100, 37238134969982100, 13513011656042074430100, 49036030210457135734021310100, 1957361459740805606124917565020990100, 937579272951542930363610919638075856505150100

Offset: 0

Jon Perry, Mar 23 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..42

Cf. A000142, A000178, A054640, A082480, A258325, A260231, A306729.

function factorial(n) {
var i,c=1;
for (i=2;i<=n;i++) c*=i;
return c;
}
a=2;
for (j=1;j<10;j++) {
a*=(factorial(j)+1);
document.write(a+", ");
}

a:= proc(n) a(n):= `if`(n=0, 2, a(n-1)*(n!+1)) end:
seq(a(n), n=0..14);  # Alois P. Heinz, May 20 2013

Table[Product[i!+1,{i,0,n}],{n,0,12}]  (* Geoffrey Critzer, May 04 2013 *)
Rest[FoldList[Times,1,Range[0,15]!+1]] (* Harvey P. Dale, May 28 2013 *)

a(n) ~ c * A000178(n), where c = A238695 = Product_{k>=0} (1 + 1/k!) = 7.364308272367257256372772509631... . - Vaclav Kotesovec, Jul 20 2015

A282529 Decimal expansion of Product_{n>=2} (1-1/n!).

Original entry on oeis.org

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

Offset: 0

Lucian Craciun, Feb 22 2017

Conjectured to be irrational, transcendental and normal, none have been shown.

			0.39533856736744556603235620043118061302665648675872...

Lucian Craciun, Table of n, a(n) for n = 0..9900
StackExchange, Closed form for Product_{n>=2} (1-1/n!), Nov 11 2016.
Wolfram Alpha, Product_{n>=2} (1-1/n!).

Cf. A238695, A247551.

evalf(product(1-1/n!, n = 2..infinity), 100); # Lucian Craciun, Feb 22 2017

RealDigits[Product[1-1/n!, {n, 2, 70}], 10, 100][[1]] (* Robert G. Wilson v, Mar 19 2014 *)

prodinf(n=3, 1-1/gamma(n)) \\ Charles R Greathouse IV, Nov 13 2014

Equals 1/A247551. - Amiram Eldar, Sep 15 2023

A269700 a(n) = floor( Product_{k>=0} (1 + n/k!) ).

Original entry on oeis.org

1, 7, 26, 69, 151, 293, 519, 862, 1361, 2062, 3019, 4297, 5969, 8121, 10848, 14261, 18481, 23646, 29908, 37437, 46419, 57061, 69586, 84242, 101297, 121042, 143793, 169893, 199710, 233642, 272117, 315592, 364560, 419545, 481109, 549849, 626403, 711448, 805703

Offset: 0

Daniel Suteu, Mar 03 2016

			For n=3, a(3) = floor(69.52294621467075235981513247145953...) = 69.

Daniel Suteu, Table of n, a(n) for n = 0..500

Cf. A238695.

Table[Floor@ Product[1 + n/k!, {k, 0, 10^2}], {n, 0, 38}] (* Michael De Vlieger, Mar 27 2016 *)

a(n) = floor(prodinf(k=0, 1+n/k!)); \\ Michel Marcus, Mar 04 2016

func a(n) {
    var (prod=1, prev=1)
    for i in (0 ..^ Inf) {
        prod *= (1 + n/i!)
        break if (prod/prev <= 1+1e-100)
        prev = prod
    }
    return floor(prod)
}
range(0, 100).each { |n| say a(n) }

A325050 a(n) = Product_{k=0..n} (k!^2 + 1).

Original entry on oeis.org

2, 4, 20, 740, 426980, 6148938980, 3187616116170980, 80970552724144881738980, 131634021973939424914920841290980, 17333817381151204925617274632152908873802980, 228254990993381085562170532497621436371926846785405002980

Offset: 0

Vaclav Kotesovec, Mar 26 2019

nonn

Cf. A055209, A101686, A217757, A238695, A325048.

Table[Product[k!^2 + 1, {k, 0, n}], {n, 0, 12}]
Table[BarnesG[n+2]^2 * Product[1 + 1/k!^2, {k, 0, n}], {n, 0, 12}]

a(n) ~ c * n^(n^2 + 2*n + 5/6) * (2*Pi)^(n+1) / (A^2 * exp(3*n^2/2 + 2*n - 1/6)), where c = Product_{k>=0} (1 + 1/k!^2) = 5.1481781945902396880952694880498895... and A is the Glaisher-Kinkelin constant A074962.

A384510 Decimal expansion of Product_{k>=0} (1 + 1/k!^2).

Original entry on oeis.org

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

Offset: 1

Kelvin Voskuijl, Aug 14 2025

			5.1481781945902396880952694880498895078630362906948...

Cf. A001044, A238695.

PARI
```
prodinf(k=0, 1 + 1/k!^2)
```

A387175 Continued fraction expansion of Product_{k>=0} (1+1/k!).

Original entry on oeis.org

7, 2, 1, 2, 1, 11, 1, 1, 2, 1, 15, 3, 1, 2, 1, 11, 1, 1, 1, 9, 1, 1, 3, 5, 9, 1, 6, 1, 1, 1, 44, 2, 1, 26, 1, 5, 1, 6, 19, 1, 1, 8, 2, 1, 8, 1, 2, 25, 9, 1, 4, 94, 1, 1, 25, 6, 2, 2, 1, 2, 4, 1, 2, 25, 28, 1, 2, 1, 2, 2, 1, 1, 4, 12, 1, 1591, 7, 1, 4, 4, 22, 1, 1, 4, 9, 1, 2, 2, 1, 2, 3, 1

Offset: 1

Kelvin Voskuijl, Aug 20 2025

			7 + 1/(2 + 1/(1 + 1/(...))) = Product_{k>=0} (1+1/k!).

Cf. A238695 (decimal expansion).

terms= 92; ContinuedFraction[Product[1+1/k!,{k,0,3*terms}],terms] (* Stefano Spezia, Aug 21 2025 *)

A386405 Decimal expansion of Product_{k>=0} (1 + 1/k!^3).

Original entry on oeis.org

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

Offset: 1

Kelvin Voskuijl, Aug 20 2025

			4.52116298976821848102535993606283488697241501987188319...

Cf. A000442, A238695, A384510.

PARI
```
prodinf(k=0, 1 + 1/k!^3)
```

cp's OEIS Frontend

A217757 Product_{i=0..n} (i! + 1).

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A282529 Decimal expansion of Product_{n>=2} (1-1/n!).

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A269700 a(n) = floor( Product_{k>=0} (1 + n/k!) ).

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A325050 a(n) = Product_{k=0..n} (k!^2 + 1).

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A384510 Decimal expansion of Product_{k>=0} (1 + 1/k!^2).

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PARI

A387175 Continued fraction expansion of Product_{k>=0} (1+1/k!).

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Mathematica

A386405 Decimal expansion of Product_{k>=0} (1 + 1/k!^3).

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