A099871 -id:A099871 - cp's OEIS frontend

A100085 Decimal expansion of Sum_{n>0} 1/(n!^n!).

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

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 08 2004

This number was called the Pomerance Number, after Carl Pomerance, in the paper by Bailey and Crandall referenced here. The paper by Martin contains a suggestion in its the Acknowledgements section by Carl Pomerance that the number might be "absolutely abnormal".

			1.250021433470507544581618655692730516577534706218865768307...

G. Harman, One hundred years of normal numbers, in M. A. Bennett et al., eds., Number Theory for the Millennium, II (Urbana, IL, 2000), 149-166, A K Peters, Natick, MA, 2002.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000.
D. H. Bailey and R. E. Crandall, On the random character of fundamental constant expansions
G. Martin, Absolutely abnormal numbers arXiv:math/0006089 [math.NT], 2000; American Mathematical Monthly 108 (October):746-754.

Cf. A073009, A099870, A099871, A099872, A099873, A100084.

RealDigits[ Sum[1/(n!)^(n!), {n, 4}], 10, 111][[1]] (* Robert G. Wilson v, Feb 26 2008 *)

suminf(n=1, 1/(n!^n!)) \\ Michel Marcus, Dec 22 2016

Edited by N. J. A. Sloane, May 16 2008 at the suggestion of R. J. Mathar

A099870 Decimal expansion of Sum_{n>0} 1/(n^log(n)).

Original entry on oeis.org

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

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Oct 29 2004

This series converges slowly. - Bernard Schott, May 23 2019

This series converges more slowly than Sum_{n>=0} 1/a^n for every a > 1 but faster than Sum_{n>=1} 1/n^p for every p > 1. - Jianing Song, Jul 25 2021

			2.23818130679669304318313699419971800961618108176500542239159050811...

J. M. Arnaudiès, P. Delezoide et H. Fraysse, Exercices résolus d'Analyse du cours de mathématiques - 2, Dunod, Exercice 9c, page 293.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A073009, A099871.

SetDefaultRealField(RealField(100)); [(&+[1/k^Log(k): k in [1..1000]])]; // G. C. Greubel, Nov 20 2018

evalf(sum(1/(n^log(n)), n=1..infinity), 110); \\ Bernard Schott, May 23 2019

s = 0; Do[s = N[s + 1/n^Log[n], 256], {n, 10^7}]; RealDigits[s, 10, 111][[1]] (* Robert G. Wilson v, Nov 02 2004 *)

default(realprecision,35);sum(n=1,50000,1./(n^log(n)))

sumpos(n=1, 1/(n^log(n))) \\ Michel Marcus, May 24 2019

numerical_approx(sum(1/k^log(k) for k in [1..1000]), digits=100) # G. C. Greubel, Nov 20 2018

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

More terms from Robert G. Wilson v, Nov 02 2004

A308915 Decimal expansion of Sum_{n>=1} 1/(log(n)^log(n)).

Original entry on oeis.org

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

Offset: 1

Bernard Schott, Jun 30 2019

This series is convergent because n^2 * 1/log(n)^log(n) = exp(log(n) * (2 - log(log(n)))) which -> 0 as n -> oo.

			6.71697061299089608814457...

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

Cf. A073009 (1/n^n), A099870 (1/n^log(n)), A099871 (1/log(n)^n).

evalf(sum(1/(log(n)^log(n)), n=1..infinity), 110);

RealDigits[N[1 + Sum[1/Log[n]^Log[n], {n, 2, Infinity}], 100]][[1]] (* Jinyuan Wang, Jul 25 2019 *)

1 + sumpos(n=2, 1/(log(n)^log(n))) \\ Michel Marcus, Jun 30 2019

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

More terms from Jon E. Schoenfield, Jun 30 2019
a(16)-a(24) from Jinyuan Wang, Jul 10 2019
More terms from Charles R Greathouse IV, Oct 21 2021

A099873 Decimal expansion of Sum_{n>=2} ((-1)^n)/(log(n)^n).

Original entry on oeis.org

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

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 02 2004

			1.52832141119264491016851348596598782062655833310082313846471081878...

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A099871.

SetDefaultRealField(RealField(100)); [(&+[(-1)^k/Log(k)^k: k in [2..1000]])]; // G. C. Greubel, Nov 20 2018

RealDigits[ Sum[ N[(-1)^n/Log[n]^n, 128], {n, 2, 160}], 10, 111][[1]] (* Robert G. Wilson v, Dec 21 2004 *)

PARI
```
sumalt(n=2,((-1)^(n))/(log(n)^n))
```

numerical_approx(sum((-1)^k/log(k)^k for k in [2..1000]), digits=100) # G. C. Greubel, Nov 20 2018

A336284 Decimal expansion of Sum_{n>=2} n^(log(n))/log(n)^n.

Original entry on oeis.org

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

Offset: 2

Bernard Schott, Jul 17 2020

This series is convergent because there exists n_1 such that for n >= n_1, n^(log(n))/log(n)^n <= (1/sqrt(e))^n.

			10.5417051152289715912697153360630929474717489965883...

Cf. A073009 (1/n^n), A099870 (1/n^log(n)), A099871 (1/log(n)^n), A308915 (1/(log(n)^log(n))).

Cf. A092605 (1/sqrt(e)).

evalf(sum(n^(log(n))/log(n)^n, n=2..infinity),100);

suminf(n=2, n^(log(n))/log(n)^n) \\ Michel Marcus, Jul 17 2020

Equals Sum_{n>=2} n^(log(n))/log(n)^n.

A336741 Decimal expansion of Sum_{n>=2} 1/log(n)^sqrt(n).

Original entry on oeis.org

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

Offset: 1

Bernard Schott, Aug 02 2020

The series u(n) = 1/log(n)^sqrt(n) is convergent because n^2 * u(n) -> 0 when n -> oo.

			4.372450021106629664550827989755553790410067553197...

J.-M. Monier, Analyse, Tome 3, 2ème année, MP.PSI.PC.PT, Dunod, 1997, Exercice 3.2.1.d p. 247.

Cf. A099870, A099871, A308915.

evalf(sum(1/(log(n)^sqrt(n), n=2..infinity), 120);

sumpos(n=2, 1/log(n)^sqrt(n)) \\ Michel Marcus, Aug 03 2020

Equals Sum_{n>=2} 1/log(n)^sqrt(n).

More terms from Jinyuan Wang, Aug 03 2020

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!.

A346670 Decimal expansion of Sum_{n>=1} 1/(n^(log(n)^2)) = Sum_{n>=1} exp(-log(n)^3).

Original entry on oeis.org

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

Offset: 1

Jianing Song, Jul 28 2021

An infinite sum that converges faster than A099870.

Note that as p > 0 gets larger and larger, the series Sum_{n>=1} 1/(n^(log(n)^p)) converges faster and faster, but will always converge more slowly than Sum_{n>=0} 1/a^n for every a > 1.

			2.07138435359817861835919830739134720946...

Jianing Song, Table of n, a(n) for n = 1..1000

Cf. A099870, A099871, A073009.

PARI
```
sumpos(n=1, 1/(n^(log(n)^2)))
```

A351687 Decimal expansion of Sum_{n>=2} (-1)^n/log(n!).

Original entry on oeis.org

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

Offset: 1

Bernard Schott, May 05 2022

This series is convergent according to the alternating series test, while series Sum_{n>=2} 1/log(n!) -> infinity (link).

			1.0769010278586314719973748207332879382948126467776416116987...

Socratic Q&A, How do you test the Series Sum_{n>=2} 1/log(n!) for convergence?

Cf. A099769 (Sum_{n>=2} (-1)^n/log(n)).

Cf. A099871, A257898, A257960, A336741.

evalf(sum((-1)^n / log(n!),n=2..infinity),120);

RealDigits[NSum[(-1)^k/Log[k!], {k, 2, Infinity}, WorkingPrecision -> 120, Method -> "AlternatingSigns"]][[1]] (* Amiram Eldar, May 05 2022 *)

sumalt(k=2, (-1)^k/log(k!)) \\ Vaclav Kotesovec, May 05 2022

Equals Sum_{k>=2} (-1)^k/log(k!).

cp's OEIS Frontend

A100085 Decimal expansion of Sum_{n>0} 1/(n!^n!).

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A099870 Decimal expansion of Sum_{n>0} 1/(n^log(n)).

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A308915 Decimal expansion of Sum_{n>=1} 1/(log(n)^log(n)).

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A099873 Decimal expansion of Sum_{n>=2} ((-1)^n)/(log(n)^n).

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A336284 Decimal expansion of Sum_{n>=2} n^(log(n))/log(n)^n.

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A336741 Decimal expansion of Sum_{n>=2} 1/log(n)^sqrt(n).

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