A306243 -id:A306243 - cp's OEIS frontend

A193424 Decimal expansion of Sum_{n>=1} log(n+1)/n!.

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

Offset: 1

N. J. A. Sloane, Aug 23 2011, based on a posting by Richard C. Schroeppel to the Math Fun Mailing List

Equal to log(2*exp(1/2*log(3*exp(1/3*log(4*exp(...)))))). - Rok Cestnik, Jan 31 2019

			1.55867043642538904...

Cf. A306243.

NSum[ Log[n + 1]/n!, {n, 1, Infinity}, WorkingPrecision -> 110, NSumTerms -> 100] // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Feb 20 2013 *)

suminf(n=1, log(n+1)/n!) \\ Michel Marcus, Jan 31 2019

More terms from Alois P. Heinz, Aug 28 2011

A385686 Decimal expansion of exp((Sum_{k>=2} log(k)/k!)/(e-1)).

Original entry on oeis.org

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

Offset: 1

Jwalin Bhatt, Jul 06 2025

The geometric mean of the Poisson distribution with parameter value 1 (A385685) approaches this constant.

			1.4210379597319607153378144890592856953982...

Cf. A296301, A306243, A382095, A385685.

N[Exp [Sum[Log[i]/Factorial[i], {i, 2, Infinity}] / (E-1) ], 120]

PARI
```
prodinf(k=2, k^(1/k!))^(1/(exp(1)-1))
```

Equals exp((Sum_{k>=2} log(k)/k!)/(e-1)).
Equals (Product_{k>=2} k^(1/k!)) ^ (1/(e-1)).
From Vaclav Kotesovec, Jul 08 2025: (Start)
Equals exp(A306243/(exp(1) - 1)).
Equals A296301^(1/(exp(1) - 1)). (End)

A336730 Decimal expansion of Sum_{n>=1} log(n)^n / n!.

Original entry on oeis.org

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

Offset: 0

Bernard Schott, Aug 02 2020

With u(n) = log(n)^n / n!, this series is convergent as u(n+1)/u(n) -> 0 when n -> oo.

			0.785672099547734935860778589192856069327...

Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année, MP, Dunod, 1997, Exercice 3.2.1.r page 279.

Cf. A099871, A306243, A308915, A336284.

evalf(sum(log(n)^n/n!,n=2..infinity),120);

suminf(n=1, log(n)^n/n!) \\ Michel Marcus, Aug 02 2020

Equals Sum_{n>=1} log(n)^n / n!.

A363767 Decimal expansion of 2^(e-2)*e^Sum_{k=2..oo} log(k)/k!.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Jun 21 2023

			3.009150722741487993563074737485316800510...

Robert A. Beeler, A Note on the number of ways to compute a determinant using cofactor expansion, Bull. Inst. Combin. Appl., 63 (2011), 36-38. [ResearchGate link]

Cf. A181044, A296301, A306243.

2^(E-2)E^NSum[Log[n]/n!, {n, 2, Infinity}, WorkingPrecision -> 110, NSumTerms -> 100] // RealDigits[#, 10, 100] &//First

Equals 2^(e-2)*e^A306243.

Equals 2^(exp(1)-2)*A296301. - Vaclav Kotesovec, Jun 22 2023

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A193424 Decimal expansion of Sum_{n>=1} log(n+1)/n!.

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A385686 Decimal expansion of exp((Sum_{k>=2} log(k)/k!)/(e-1)).

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A336730 Decimal expansion of Sum_{n>=1} log(n)^n / n!.

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A363767 Decimal expansion of 2^(e-2)*e^Sum_{k=2..oo} log(k)/k!.

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