A296301 -id:A296301 - cp's OEIS frontend

A306243 Decimal expansion of Sum_{n>=2} log(n)/n!.

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

Offset: 0

Rok Cestnik, Jan 31 2019

			0.6037828627914879884...

István Mező, Problem 11806, Problems and Solutions, The American Mathematical Monthly, Vol. 121, No. 10 (2014), p. 947; Parseval and Kummer, Solution to Problem 11806 by Omran Kouba, ibid., Vol. 123, No. 9 (2016), pp. 943-944.

Cf. A001620, A061444, A193424, A296301, A336730.

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

suminf(n=2, log(n)/n!) \\ Michel Marcus, Jan 31 2019

Equal to log(exp(1/2*log(2*exp(1/3*log(3*exp(1/4*log(4*exp(...)))))))).
Equals log(A296301). - Vaclav Kotesovec, Jun 22 2023
Equals Integral_{x=0..2*Pi} log(Gamma(x/(2*Pi))) * exp(cos(x)) * sin(x + sin(x)) dx - (e-1)*(log(2*Pi)+gamma), where gamma is Euler's constant (A001620) (Mező, 2014). - Amiram Eldar, Jan 25 2024
Equals Integral_{x=0..1} (exp(x) - 1)/(x*log(x)) - (exp(1) - 1)/log(x) dx. - Velin Yanev, Nov 29 2024

A296547 Decimal expansion of (2 + (3 + (4 + (5 + ...)^(1/5))^(1/4))^(1/3))^(1/2).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Dec 15 2017

			1.9116392162458242838833095376406246646085815394...

Eric Weisstein's World of Mathematics, Nested Radical

Cf. A072449, A296301.

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)

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

cp's OEIS Frontend

A306243 Decimal expansion of Sum_{n>=2} log(n)/n!.

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A296547 Decimal expansion of (2 + (3 + (4 + (5 + ...)^(1/5))^(1/4))^(1/3))^(1/2).

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

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

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