A104139 -id:A104139 - cp's OEIS frontend

A007524 Decimal expansion of log_10(2).

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

Offset: 0

N. J. A. Sloane

Log_10(2) is the probability that 1 be first significant digit occurring in data collections (Benford's law). - Lekraj Beedassy, Jan 21 2005
When adding two sound power sources of x decibels, the resulting sound power is x + 10*log_10(2), that is x + 3.01... decibels. - Jean-François Alcover, Jun 21 2013
In engineering (all branches, but particularly electronic and electrical) power and amplitude ratios are measured rigorously in decibels (dB). This constant, with offset 1 (i.e., 3.01... = 10*A007524) is the dB equivalent of a 2:1 power ratio or, equivalently, sqrt(2):1 amplitude ratio. - Stanislav Sykora, Dec 11 2013

			0.3010299956639811952137388947244930267681898814621085413104274611271...

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 24, 249.
T. Hill, "Manipulation, or the First Significant Numeral Determines the Law", in 'La Recherche', No. 2 1999 pp. 72-76 (or No. 116 1999 pp. 72-75), Paris.
M. E. Lines, A Number For Your Thought, pp. 43-52 Institute of Physics Pub. London 1990.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
I. Stewart, L'univers des nombres, "1 est plus probable que 9", pp. 57-61, Belin-Pour La Science, Paris 2000.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 27.

Harry J. Smith, Table of n, a(n) for n = 0..20000
K. Brown, Benford's Law.
C. K. Caldwell, The Prime Glossary, Benford's law.
I. Gent and T. Walsh, Benford's Law.
T. P. Hill, The first digital phenomenon. [Broken link]
T. P. Hill, The First-Digit Phenomenon.
T. P. Hill, The First-Digit Phenomenon (Accompanying Diagrams).
R. Matthews, The Power of One.
S. J. Miller, Some Thoughts on Benford's Law.
M. J. Nigrini, Benford's Law. [Broken link]
I. Peterson, Mathtrek, First Digits. [Broken link]
L. Pietronero et al., The Uneven Distribution of Numbers in Nature, arXiv:cond-mat/9808305 [cond-mat.stat-mech], 1998.
Simon Plouffe, The LOG of 2(in base 10).
J. Walthoe, Looking out for number one.
Eric Weisstein's World of Mathematics, Benford's Law.
Eric Weisstein's World of Mathematics, Mersenne Number.
Wikipedia, Benford's law.
Wikipedia, Decibel.
Index entries for sequences related to Benford's law.
Index entries for transcendental numbers.

Cf. decimal expansion of log_10(m): this sequence, A114490 (m = 3), A114493 (m = 4), A153268 (m = 5), A153496 (m = 6), A153620 (m = 7), A153790 (m = 8), A104139 (m = 9), A154182 (m = 11), A154203 (m = 12), A154368 (m = 13), A154478 (m = 14), A154580 (m = 15), A154794 (m = 16), A154860 (m = 17), A154953 (m = 18), A155062 (m = 19), A155522 (m = 20), A155677 (m = 21), A155746 (m = 22), A155830 (m = 23), A155979 (m = 24).

RealDigits[Log[10, 2], 10, 120][[1]] (* Harvey P. Dale, Dec 19 2011 *)

default(realprecision, 20080); x=log(2)/log(10); d=0; for (n=0, 20000, x=(x-d)*10; d=floor(x); write("b007524.txt", n, " ", d)); \\ Harry J. Smith, Apr 15 2009

log_10(2) = log(2)/log(10) = log(2)/(log(2) + log(5)).

Equals 1/A020862. - R. J. Mathar, Jul 31 2025

Definition corrected by Franklin T. Adams-Watters, Apr 13 2006

Final digits of sequence corrected using the b-file. - N. J. A. Sloane, Aug 30 2009

A038772 Numbers not divisible by any of their digits.

Original entry on oeis.org

23, 27, 29, 34, 37, 38, 43, 46, 47, 49, 53, 54, 56, 57, 58, 59, 67, 68, 69, 73, 74, 76, 78, 79, 83, 86, 87, 89, 94, 97, 98, 203, 207, 209, 223, 227, 229, 233, 239, 247, 249, 253, 257, 259, 263, 267, 269, 277, 283, 289, 293, 299, 307, 308, 323, 329, 334, 337, 338

Offset: 1

Henry Bottomley, May 04 2000

A038769(a(n)) = 0; complement of A038770.
This is a regular language when written in decimal, though the minimal regular expression is probably thousands of characters long. - Charles R Greathouse IV, Aug 19 2011
Exponential density 0.954... = A104139. Asymptotically 8/35 * n^0.954... + O(n^0.903...) members up to n. - Charles R Greathouse IV, Jul 22 2012

			34 is divisible by neither 3 nor 4.
35 is excluded because 5 is a divisor of 35, but 37 is included because neither 3 nor 7 is a divisor of 37

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences

Cf. A327561 (counts), A038770 (complement).

Cf. also A034709, A034837, A038769.

import Data.Char (digitToInt)
a038772 n = a038772_list !! (n-1)
a038772_list = filter p [1..] where
   p n = all (> 0) $ map ((mod n) . digitToInt) $ filter (> '0') $ show n
-- Reinhard Zumkeller, Jun 19 2011

[k:k in [1..340]| forall{c:c in Set(Intseq(k)) diff {0}|k mod c ne 0}]; // Marius A. Burtea, Dec 22 2019

nddQ[n_]:=Module[{idn=DeleteCases[IntegerDigits[n],0]},And@@Table[ !Divisible[n, idn[[i]]],{i,Length[idn]}]]; Select[Range[350],nddQ] (* Harvey P. Dale, Nov 01 2011 *)

is(n)=my(v=vecsort(eval(Vec(Str(n))), , 8)); for(i=if(v[1], 1, 2), #v, if(n%v[i]==0, return(0))); 1 \\ Charles R Greathouse IV, Jul 22 2011

def ok(n): return not any(n%int(d) == 0 for d in str(n) if d != '0')
print(list(filter(ok, range(1, 339)))) # Michael S. Branicky, May 20 2021

Edited by N. J. A. Sloane, Nov 17 2008 at the suggestion of R. J. Mathar

A154580 Decimal expansion of log_10 (15).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 30 2009

			1.1760912590556812420812890085306222824319389827285873235194...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for transcendental numbers

Cf. decimal expansion of log_10(m): A007524 (m=2), A114490 (m=3), A114493 (m=4), A153268 (m=5), A153496 (m=6), A153620 (m=7), A153790 (m=8), A104139 (m=9), A154182 (m=11), A154203 (m=12), A154368 (m=13), A154478 (m=14), this sequence, A154794 (m=16), A154860 (m=17), A154953 (m=18), A155062 (m=19), A155522 (m=20), A155677 (m=21), A155746 (m=22), A155830 (m=23), A155979 (m=24).

RealDigits[Log[10, 15], 10, 100][[1]] (* Vincenzo Librandi, Sep 11 2013 *)

log(15)/log(10) \\ Charles R Greathouse IV, Mar 17 2018

Equals A016638 / A002392 = (1+A152914)/(1+A152675). - R. J. Mathar, Jul 29 2024

A114490 Decimal expansion of log_10(3).

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Dec 01 2005

			0.477121254...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Motzkin Number
Index entries for transcendental numbers

Cf. A114473.

Cf. decimal expansion of log_10(m): A007524 (m=2), this sequence, A114493 (m=4), A153268 (m=5), A153496 (m=6), A153620 (m=7), A153790 (m=8), A104139 (m=9), A154182 (m=11), A154203 (m=12), A154368 (m=13), A154478 (m=14), A154580 (m=15), A154794 (m=16), A154860 (m=17), A154953 (m=18), A155062 (m=19), A155522 (m=20), A155677 (m=21), A155746 (m=22), A155830 (m=23), A155979 (m=24).

RealDigits[Log[10, 3], 10, 100][[1]] (* Vincenzo Librandi, Sep 10 2013 *)

log(3)/log(10) \\ Michel Marcus, Jan 25 2015

Equals 1/A152566. - R. J. Mathar, Jul 31 2025

A154203 Decimal expansion of log_10 (12).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 30 2009

			1.0791812460476248277225056927041013627365086271149129474507...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for transcendental numbers

Cf. decimal expansion of log_10(m): A007524 (m=2), A114490 (m=3), A114493 (m=4), A153268 (m=5), A153496 (m=6), A153620 (m=7), A153790 (m=8), A104139 (m=9), A154182 (m=11), this sequence, A154368 (m=13), A154478 (m=14), A154580 (m=15), A154794 (m=16), A154860 (m=17), A154953 (m=18), A155062 (m=19), A155522 (m=20), A155677 (m=21), A155746 (m=22), A155830 (m=23), A155979 (m=24).

RealDigits[Log[10,12],10,100][[1]] (* Vincenzo Librandi,  Sep 11 2013 *)

log(12)/log(10) \\ Charles R Greathouse IV, Aug 04 2020

Equals A007524 + A153596. - R. J. Mathar, Jan 07 2021

A114493 Decimal expansion of log_10(4).

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Dec 01 2005

In engineering (all branches, but particularly electronic and electrical) power and amplitude ratios are measured rigorously in decibels (dB). This constant, with offset 1 (i.e., 6.02... = 10*A114493) is the dB equivalent of a 2:1 amplitude ratio or, equivalently, 4:1 power ratio. - Stanislav Sykora, Dec 11 2013

			0.602059991...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Catalan Number
Eric Weisstein's World of Mathematics, Central Binomial Coefficient
Wikipedia, Decibel
Index entries for transcendental numbers

Cf. A000108, A114466.

Cf. decimal expansion of log_10(m): A007524 (m=2), A114490 (m=3), this sequence, A153268 (m=5), A153496 (m=6), A153620 (m=7), A153790 (m=8), A104139 (m=9), A154182 (m=11), A154203 (m=12), A154368 (m=13), A154478 (m=14), A154580 (m=15), A154794 (m=16), A154860 (m=17), A154953 (m=18), A155062 (m=19), A155522 (m=20), A155677 (m=21), A155746 (m=22), A155830 (m=23), A155979 (m=24).

log10(4.0) ; # R. J. Mathar, Feb 21 2013

RealDigits[Log[10, 4], 10, 100][[1]] (* Vincenzo Librandi, Sep 10 2013 *)

log(4)/log(10) \\ Charles R Greathouse IV, Aug 04 2020

A016627 divided by A002392. Two times A007524. - R. J. Mathar, Feb 21 2013

Equals 1/A154155. - R. J. Mathar, Jul 31 2025

A153620 Decimal expansion of log_10 (7).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Oct 30 2009

			.84509804001425683071221625859263619348357239632396540650363...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for transcendental numbers

Cf. decimal expansion of log_10(m): A007524 (m=2), A114490 (m=3), A114493 (m=4), A153268 (m=5), A153496 (m=6), this sequence, A153790 (m=8), A104139 (m=9), A154182 (m=11), A154203 (m=12), A154368 (m=13), A154478 (m=14), A154580 (m=15), A154794 (m=16), A154860 (m=17), A154953 (m=18), A155062 (m=19), A155522 (m=20), A155677 (m=21), A155746 (m=22), A155830 (m=23), A155979 (m=24).

RealDigits[Log[10, 7], 10, 100][[1]] (* Vincenzo Librandi, Sep 10 2013 *)

log(7)/log(10) \\ Charles R Greathouse IV, Aug 04 2020

A153496 Decimal expansion of log_10 (6).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Oct 30 2009

			.77815125038364363250876679797960833596831874565280440614029...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for transcendental numbers

Cf. decimal expansion of log_10(m): A007524 (m=2), A114490 (m=3), A114493 (m=4), A153268 (m=5), this sequence, A153620 (m=7), A153790 (m=8), A104139 (m=9), A154182 (m=11), A154203 (m=12), A154368 (m=13), A154478 (m=14), A154580 (m=15), A154794 (m=16), A154860 (m=17), A154953 (m=18), A155062 (m=19), A155522 (m=20), A155677 (m=21), A155746 (m=22), A155830 (m=23), A155979 (m=24).

RealDigits[Log[10, 6], 10, 100][[1]] (* Vincenzo Librandi, Sep 10 2013 *)

log(6)/log(10) \\ Charles R Greathouse IV, Aug 04 2020

Equals 1/A154157. - R. J. Mathar, Jul 31 2025

A154182 Decimal expansion of log_10 (11).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 30 2009

			1.0413926851582250407501999712430242417067021904664530945965...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for transcendental numbers

Cf. decimal expansion of log_10(m): A007524 (m=2), A114490 (m=3), A114493 (m=4), A153268 (m=5), A153496 (m=6), A153620 (m=7), A153790 (m=8), A104139 (m=9), this sequence, A154203 (m=12), A154368 (m=13), A154478 (m=14), A154580 (m=15), A154794 (m=16), A154860 (m=17), A154953 (m=18), A155062 (m=19), A155522 (m=20), A155677 (m=21), A155746 (m=22), A155830 (m=23), A155979 (m=24).

RealDigits[Log[10, 11], 10, 100][[1]] (* Vincenzo Librandi, Sep 10 2013 *)

log(11)/log(10) \\ Charles R Greathouse IV, Aug 04 2020

A153268 Decimal expansion of log_10 (5).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Oct 30 2009

			.69897000433601880478626110527550697323181011853789145868957...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for transcendental numbers

Cf. decimal expansion of log_10(m): A007524 (m=2), A114490 (m=3), A114493 (m=4), this sequence, A153496 (m=6), A153620 (m=7), A153790 (m=8), A104139 (m=9), A154182 (m=11), A154203 (m=12), A154368 (m=13), A154478 (m=14), A154580 (m=15), A154794 (m=16), A154860 (m=17), A154953 (m=18), A155062 (m=19), A155522 (m=20), A155677 (m=21), A155746 (m=22), A155830 (m=23), A155979 (m=24).

RealDigits[Log[10, 5], 10, 100][[1]] (* Vincenzo Librandi, Sep 10 2013 *)

log(5)/log(10) \\ Charles R Greathouse IV, Aug 04 2020

Equals 1/A154156. - R. J. Mathar, Jul 31 2025

cp's OEIS Frontend

A007524 Decimal expansion of log_10(2).

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A038772 Numbers not divisible by any of their digits.

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A154580 Decimal expansion of log_10 (15).

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A114490 Decimal expansion of log_10(3).

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A154203 Decimal expansion of log_10 (12).

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A114493 Decimal expansion of log_10(4).

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A153620 Decimal expansion of log_10 (7).

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