A053511 -id:A053511 - cp's OEIS frontend

A053510 Decimal expansion of log(Pi).

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

Offset: 1

Hsu, Po-Wei (Benny) (arsene_lupin(AT)intekom.co.za), Jan 14 2000

Also the least positive x such that sin(exp(x))==0.
Also real part of log(log(-1)). - Stanislav Sykora, May 11 2015
Cheng, Dietel, Herblot, Huang, Krieger, Marques, Mason, Mereb, & Wilson show, expanding a remark by S. Lang, that Schanuel's conjecture implies that this constant and Pi are algebraically independent over a set E which includes the algebraic numbers and (in a technical sense) allows any finite number of exponentiations, see the paper for details and a still more general result. - Charles R Greathouse IV, Dec 15 2019

			1.1447298858494001741...

Wolfram Research, 1991 Mathematica Conference, Elementary Tutorial Notes, Section 1, Introduction to Mathematica, Paul Abbott, page 25.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Chuangxun Cheng, Brian Dietel, Mathilde Herblot, Jingjing Huang, Holly Krieger, Diego Marques, Jonathan Mason, Martin Mereb, S. Robert Wilson, Some consequences of Schanuel's conjecture, Journal of Number Theory 129:6 (2009), pp. 1464-1467.
Michael Penn, Frullani Integral, YouTube video, 2021.

Cf. A000796, A053511.

R:= RealField(100); Log(Pi(R)); // G. C. Greubel, May 15 2019

Mathematica
```
RealDigits[Log[Pi], 10, 111][[1]]
```

log(Pi) \\ Charles R Greathouse IV, Jan 04 2016

numerical_approx(log(pi), digits=100) # G. C. Greubel, May 15 2019

Equals log(log(-1)) - (Pi/2)*I. - Stanislav Sykora, May 11 2015
Equals 1 + Sum_{n>=1} zeta(2*n)/(n*(2*n+1)*2^(2*n)), where zeta is the Riemann zeta function. - Vaclav Kotesovec, Mar 04 2016
Equals 3/2 - Sum_{k>=1} (zeta(2*k)-1)/(k+1). - Vaclav Kotesovec, Jun 19 2021

More terms from James Sellers, Jan 20 2000

A216582 Decimal expansion of the logarithm of Pi to base 2.

Original entry on oeis.org

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

Offset: 1

Alonso del Arte, Sep 09 2012

			1.651496129472318798...

Jörg Arndt and Christoph Haenel, Pi Unleashed, New York: Springer (2001), p. 239.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A002162, A020857, A053510, A053511, A061444, A094642.

SetDefaultRealField(RealField(100)); R:= RealField(); Log(Pi(R))/Log(2); // G. C. Greubel, Apr 08 2019

evalf(log[2](Pi)) ; # R. J. Mathar, Sep 11 2012

Mathematica
```
RealDigits[Log[2, Pi], 10, 105][[1]]
```

log(Pi)/log(2) \\ Michel Marcus, Mar 05 2019

log(pi,2).n(digits=100) # Jani Melik, Oct 05 2012

Log_2(Pi) = log(Pi) / log(2) = A053510 / A002162.

Equals (A061444 / A002162) - 1 = (A094642 / A002162) + 1. - John W. Nicholson, Mar 12 2019

A219723 Numerators of convergents to log_10(Pi).

Original entry on oeis.org

0, 1, 87, 349, 436, 785, 1221, 5669, 296009, 597687, 893696, 48857271, 98608238, 936331413, 1034939651, 3006210715, 4041150366, 11088511447, 37306684707, 458768727931, 496075412638, 954844140569, 2405763693776, 5766371528121, 8172135221897

Offset: 1

Arkadiusz Wesolowski, Nov 26 2012

			0, 1/2, 87/175, 349/702, 436/877, 785/1579, 1221/2456, 5669/11403, ....

Eric Weisstein's World of Mathematics, Pi

Cf. A053511 (decimal), A219724 (denominators).

Mathematica
```
Numerator@Convergents[Log[10, Pi], 25]
```

A219724 Denominators of convergents to log_10(Pi).

Original entry on oeis.org

1, 2, 175, 702, 877, 1579, 2456, 11403, 595412, 1202227, 1797639, 98274733, 198347105, 1883398678, 2081745783, 6046890244, 8128636027, 22304162298, 75041122921, 922797637350, 997838760271, 1920636397621, 4839111555513, 11598859508647, 16437971064160

Offset: 1

Arkadiusz Wesolowski, Nov 26 2012

Pi^a(n) gets increasingly close to 10^A219723(n).

			0, 1/2, 87/175, 349/702, 436/877, 785/1579, 1221/2456, 5669/11403, ....

Eric Weisstein's World of Mathematics, Pi

Cf. A053511 (decimal), A219723 (numerators).

Denominator@Convergents[Log[10, Pi], 25]

A386725 a(n) is the nearest integer to n/log_10(Pi).

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129

Offset: 0

Stefano Spezia, Jul 31 2025

Differs from A005843 for n > 43.

For n > 0, a(n) is the nonnegative integer such that abs(10^(n/a(n)) - Pi) is minimum.

			a(n) = 2*n corresponds to the Brahmagupta's approximation 10^(1/2) = sqrt(10) of Pi (cf. A010467).

Cf. A000796, A002160, A005843, A010467, A053511, A385570.

a[n_]:=Round[n/Log10[Pi]]; Array[a,65,0]

a(n) = round(n/log10(Pi)).

a(n) >= 2*n.

A073365 Decimal expansion of log(log(Pi)).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Jul 29 2002

Cheng, Dietel, Herblot, Huang, Krieger, Marques, Mason, Mereb, & Wilson show, expanding a remark by S. Lang, that Schanuel's conjecture implies that this constant and Pi are algebraically independent over a set E which includes the algebraic numbers and (in a technical sense) allows any finite number of exponentiations, see the paper for details and a still more general result. - Charles R Greathouse IV, Dec 16 2019

			0.13516870162052962769995812823...

Chuangxun Cheng, Brian Dietel, Mathilde Herblot, Jingjing Huang, Holly Krieger, Diego Marques, Jonathan Mason, Martin Mereb, S. Robert Wilson, Some consequences of Schanuel's conjecture, Journal of Number Theory 129:6 (2009), pp. 1464-1467.

Cf. A000796 (Pi), A053510 (log(Pi)), A053511 (log_10(Pi)).

RealDigits[Log[Log[Pi]],10,120][[1]] (* Harvey P. Dale, Mar 11 2017 *)

PARI
```
log(log(Pi))
```

A235955 Decimal expansion of log_Pi 10.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jun 13 2014

			= 2.01146586758806093876472204728870869669458302073723988728116...

Cf. A053511.

Mathematica
```
RealDigits[ Log[ Pi, 10], 10, 111]
```

1/A053511.

A348960 a(n) = floor(log_10(Pi*n!)).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 20, 21, 22, 24, 25, 27, 28, 29, 31, 32, 34, 35, 37, 38, 40, 42, 43, 45, 46, 48, 50, 51, 53, 54, 56, 58, 59, 61, 63, 64, 66, 68, 70, 71, 73, 75, 77, 78, 80, 82, 84, 85, 87, 89, 91, 93, 95, 96, 98, 100

Offset: 0

Paul F. Marrero Romero, Nov 05 2021

nonn

Cf. A004216, A034886, A053511.

Array[Floor@ Log10[Pi*#!] &, 71, 0] (* Michael De Vlieger, Nov 05 2021 *)

a(n) = floor(log_10(Pi*n!)).

a(n) = floor(A053511 + log_10(n!)).

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A053510 Decimal expansion of log(Pi).

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A216582 Decimal expansion of the logarithm of Pi to base 2.

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A219723 Numerators of convergents to log_10(Pi).

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A219724 Denominators of convergents to log_10(Pi).

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A386725 a(n) is the nearest integer to n/log_10(Pi).

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A073365 Decimal expansion of log(log(Pi)).

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Mathematica

PARI

A235955 Decimal expansion of log_Pi 10.

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