A002392 -id:A002392 - cp's OEIS frontend

A154580 Decimal expansion of log_10 (15).

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

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

A104139 Decimal expansion of log_10(9).

Original entry on oeis.org

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

Offset: 0

Lekraj Beedassy, Mar 07 2005

			log_10(9) = 0.95424250943932487459005580651...

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), A153620 (m = 7), A153790 (m = 8), this sequence, 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[N[Log[10, 9], 150]] (* Stefan Steinerberger, Mar 14 2006 *)

log(9)/log(10) \\ R. J. Mathar, Mar 11 2008

Equals A016632 / A002392 . - R. J. Mathar, Mar 11 2008

More terms from Stefan Steinerberger, Mar 14 2006

More terms from R. J. Mathar, Mar 11 2008

A154158 Decimal expansion of log_7(10).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 30 2009

			1.1832946624549383268179285616468591481654445229423947233563...

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

Cf. decimal expansion of log_7(m): A152713 (m=2), A152945 (m=3), A153103 (m=4), A153203 (m=5), A153463 (m=6), A153755 (m=8), A113211 (m=9), this sequence, A154179 (m=11), A154200 (m=12), A154294 (m=13), A154467 (m=14), A154572 (m=15), A154793 (m=16), A154857 (m=17), A154912 (m=18), A155059 (m=19), A155496 (m=20), A155591 (m=21), A155735 (m=22), A155824 (m=23), A155964 (m=24).

SetDefaultRealField(RealField(100)); Log(10)/Log(7); // G. C. Greubel, Sep 02 2018

RealDigits[Log[7, 10], 10,100][[1]] (* Vincenzo Librandi, Aug 31 2013 *)

default(realprecision, 100); log(10)/log(7) \\ G. C. Greubel, Sep 02 2018

Equals A002392 / A016630 = 1/A153620. - R. J. Mathar, Jul 31 2025

A083910 Number of divisors of n that are congruent to 0 modulo 10.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0

Offset: 1

Reinhard Zumkeller, May 08 2003

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A000005, A000007, A001227, A010879, A027750, A183063.
Cf. A001620, A002392.
Cf. A083911, A083912, A083913, A083914, A083915, A083916, A083917, A083918, A083919.

a083910 = sum . map (a000007 . a010879) . a027750_row
-- Reinhard Zumkeller, Jan 15 2013

ndc10[n_]:=Count[Divisors[n],?(Divisible[#,10]&)]; Array[ndc10,110] (* _Harvey P. Dale, Jan 05 2013 *)
a[n_] := If[Divisible[n, 10], DivisorSigma[0, n/10], 0]; Array[a, 100] (* Amiram Eldar, Dec 30 2023 *)

a(n)=if(n%10,0,numdiv(n/10)) \\ Charles R Greathouse IV, Sep 27 2015

a(n) = A000005(n) - A083911(n) - A083912(n) - A083913(n) - A083914(n) - A083915(n) - A083916(n) - A083917(n) - A083918(n) - A083919(n).
a(10k) = tau(k) = A000005(k); a(n) = 0 if 10 does not divide n. - Franklin T. Adams-Watters, Apr 15 2007
G.f.: Sum_{k>=1} x^(10*k)/(1 - x^(10*k)). - Ilya Gutkovskiy, Sep 11 2019
Sum_{k=1..n} a(k) = n*log(n)/10 + c*n + O(n^(1/3)*log(n)), where c = (2*gamma - 1 - log(10))/10 = -0.214815..., and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Dec 30 2023

A083911 Number of divisors of n that are congruent to 1 modulo 10.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2

Offset: 1

Reinhard Zumkeller, May 08 2003

Amiram Eldar, Table of n, a(n) for n = 1..10000
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A000005, A001227, A010879.
Cf. A001620, A002392, A306716 (psi(1/10)).
Cf. A083910, A083912, A083913, A083914, A083915, A083916, A083917, A083918, A083919.

a[n_] := DivisorSum[n, 1 &, Mod[#, 10] == 1 &]; Array[a, 100] (* Amiram Eldar, Dec 30 2023 *)

a(n) = sumdiv(n, d, d % 10 == 1); \\ Amiram Eldar, Dec 30 2023

a(n) = A000005(n) - A083910(n) - A083912(n) - A083913(n) - A083914(n) - A083915(n) - A083916(n) - A083917(n) - A083918(n) - A083919(n).
G.f.: Sum_{k>=1} x^k/(1 - x^(10*k)). - Ilya Gutkovskiy, Sep 11 2019
Sum_{k=1..n} a(k) = n*log(n)/10 + c*n + O(n^(1/3)*log(n)), where c = gamma(1,10) - (1 - gamma)/10 = 0.769838..., gamma(1,6) = -(psi(1/10) + log(10))/10 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Dec 30 2023

A083913 Number of divisors of n that are congruent to 3 modulo 10.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 2, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 1, 2, 0, 0, 2, 0, 0, 1, 1, 0, 1, 0, 0, 2, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 2, 0, 0, 1, 0, 0, 2, 0, 0, 1, 1, 1, 1

Offset: 1

Reinhard Zumkeller, May 08 2003

Amiram Eldar, Table of n, a(n) for n = 1..10000
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A000005, A001227, A010879.
Cf. A001620, A002392, A354642 (psi(3/10)).
Cf. A083910, A083911, A083912, A083914, A083915, A083916, A083917, A083918, A083919.

a[n_] := DivisorSum[n, 1 &, Mod[#, 10] == 3 &]; Array[a, 100] (* Amiram Eldar, Dec 30 2023 *)

a(n) = sumdiv(n, d, d % 10 == 3); \\ Amiram Eldar, Dec 30 2023

a(n) = A000005(n) - A083910(n) - A083911(n) - A083912(n) - A083914(n) - A083915(n) - A083916(n) - A083917(n) - A083918(n) - A083919(n).
G.f.: Sum_{k>=1} x^(3*k)/(1 - x^(10*k)). - Ilya Gutkovskiy, Sep 11 2019
Sum_{k=1..n} a(k) = n*log(n)/10 + c*n + O(n^(1/3)*log(n)), where c = gamma(3,10) - (1 - gamma)/10 = 0.07771547..., gamma(3,10) = -(psi(3/10) + log(10))/10 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Dec 30 2023

A083917 Number of divisors of n that are congruent to 7 modulo 10.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1

Offset: 1

Reinhard Zumkeller, May 08 2003

Amiram Eldar, Table of n, a(n) for n = 1..10000
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A000005, A001227, A010879.
Cf. A001620, A002392, A354643 (psi(7/10)).
Cf. A083910, A083911, A083912, A083913, A083914, A083915, A083916, A083918, A083919.

Table[Count[Divisors[n],?(Mod[#,10]==7&)],{n,110}] (* _Harvey P. Dale, Sep 15 2023 *)
a[n_] := DivisorSum[n, 1 &, Mod[#, 10] == 7 &]; Array[a, 100] (* Amiram Eldar, Dec 30 2023 *)

a(n) = sumdiv(n, d, d % 10 == 7); \\ Amiram Eldar, Dec 30 2023

a(n) = A000005(n) - A083910(n) - A083911(n) - A083912(n) - A083913(n) - A083914(n) - A083915(n) - A083916(n) - A083918(n) - A083919(n).
G.f.: Sum_{k>=1} x^(7*k)/(1 - x^(10*k)). - Ilya Gutkovskiy, Sep 11 2019
Sum_{k=1..n} a(k) = n*log(n)/10 + c*n + O(n^(1/3)*log(n)), where c = gamma(7,10) - (1 - gamma)/10 = -0.150534..., gamma(7,10) = -(psi(7/10) + log(10))/10 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Dec 30 2023

A083919 Number of divisors of n that are congruent to 9 modulo 10.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0

Offset: 1

Reinhard Zumkeller, May 08 2003

Amiram Eldar, Table of n, a(n) for n = 1..10000
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A000005, A001227, A010879.
Cf. A001620, A002392, A354644 (psi(9/10)).
Cf. A083910, A083911, A083912, A083913, A083914, A083915, A083916, A083917, A083918.

a[n_] := DivisorSum[n, 1 &, Mod[#, 10] == 9 &]; Array[a, 100] (* Amiram Eldar, Dec 30 2023 *)

a(n) = sumdiv(n, d, d % 10 == 9); \\ Amiram Eldar, Dec 30 2023

a(n) = A000005(n) - A083910(n) - A083911(n) - A083912(n) - A083913(n) - A083914(n) - A083915(n) - A083916(n) - A083917(n) - A083918(n).
G.f.: Sum_{k>=1} x^(9*k)/(1 - x^(10*k)). - Ilya Gutkovskiy, Sep 11 2019
Sum_{k=1..n} a(k) = n*log(n)/10 + c*n + O(n^(1/3)*log(n)), where c = gamma(9,10) - (1 - gamma)/10 = -0.197044..., gamma(9,10) = -(psi(9/10) + log(10))/10 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Dec 30 2023

A002285 Decimal expansion of common logarithm of e.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Sometimes also called Briggs's constant after the English mathematician Henry Briggs (1561-1630). - Martin Renner, Jan 03 2022

			0.4342944819...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 25, equations 25:14:4 at page 232.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 27.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Vladimir Ivanovich Smirnov, A course of higher mathematics, vol. 1 , Pergamon Press, 1964, p. 79.
Jonathan Sondow and Eric W. Weisstein, MathWorld: Harmonic Number.
Horace S. Uhler, Recalculation and extension of the modulus and of the logarithms of 2, 3, 5, 7 and 17, Proc. Nat. Acad. Sci. U.S.A. 26, (1940), pp. 205-212.
Wikipedia, Henry Briggs.
Index entries for transcendental numbers.

Cf. A001008, A002392, A002805, A114467, A114468.

evalf[100](1/log(10)); # Martin Renner, Jan 03 2022

RealDigits[N[1/Log[10], 100]][[1]] (* Vincenzo Librandi, Mar 25 2013 *)
RealDigits[Log10[E],10,120][[1]] (* Harvey P. Dale, Apr 17 2022 *)

1/log(10) \\ Charles R Greathouse IV, Jan 04 2016

Equals log_10(e) = 1/log(10) = 1/A002392. - Eric Desbiaux, Jun 27 2009

Conjecture by Eric Weisstein: Equals lim_{n->oo} b(n)/10^(n-1), for b=A114467 or b=A114468 (i.e., is the limit of the decimal expansion of the number of decimal digits in both the numerator and denominator of the (10^n)th harmonic number). More generally, log_k(e) seems to equal lim_{n->oo} floor(log_k(b(k^n)))/k^(n-1), for b=A001008 or b=A002805 and k >= 2. - Natalia L. Skirrow, Feb 12 2023

cp's OEIS Frontend

A154580 Decimal expansion of log_10 (15).

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A114493 Decimal expansion of log_10(4).

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Maple

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A104139 Decimal expansion of log_10(9).

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A154158 Decimal expansion of log_7(10).

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Magma

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A083910 Number of divisors of n that are congruent to 0 modulo 10.

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Haskell

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A083911 Number of divisors of n that are congruent to 1 modulo 10.

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A083913 Number of divisors of n that are congruent to 3 modulo 10.

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