A099870 -id:A099870 - 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

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

A100084 Decimal expansion of Sum_{n>0} 1/(n^(n!)).

Original entry on oeis.org

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

Offset: 1

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

			1.251371742112486405937272764835634431067154077181927...

Cf. A073009, A099870 to A099873.

PARI
```
sum(n=1,9,1/(n^(n!)),0.)
```

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

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)))
```

A336987 Decimal expansion of Sum_{n>=2} sqrt(n)^log(n)/log(n)^sqrt(n).

Original entry on oeis.org

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

Offset: 2

Bernard Schott, Aug 10 2020

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

			32.219419584243365152435936117722884...

J. Moisan & A. Vernotte, Analyse, Topologie et Séries, Exercices corrigés de Mathématiques Spéciales, Ellipses, 1991, Exercice B-1 a-3 pp. 70, 87-88.

Cf. A099870, A308915, A336284, A336741.

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

default(realprecision, 100); sumpos(n=2, sqrt(n)^log(n)/log(n)^sqrt(n)) \\ Michel Marcus, Aug 10 2020

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

a(37)-a(101) from Robert Price, Aug 21 2020

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A100084 Decimal expansion of Sum_{n>0} 1/(n^(n!)).

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Maple

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A336987 Decimal expansion of Sum_{n>=2} sqrt(n)^log(n)/log(n)^sqrt(n).

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Maple