A002392 -id:A002392 - cp's OEIS frontend

A083912 Number of divisors of n that are congruent to 2 modulo 10.

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

Offset: 1

Reinhard Zumkeller, May 08 2003

Antti Karttunen, Table of n, a(n) for n = 1..65537
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, A200135 (psi(1/5)).
Cf. A083910, A083911, A083913, A083914, A083915, A083916, A083917, A083918, A083919.

a[n_] := Sum[If[Mod[d, 10] == 2, 1, 0], {d, Divisors[n]}];
Array[a, 105] (* Jean-François Alcover, Dec 02 2021 *)
a[n_] := DivisorSum[n, 1 &, Mod[#, 10] == 2 &]; Array[a, 100] (* Amiram Eldar, Dec 30 2023 *)

A083912(n) = sumdiv(n,d,2==(d%10)); \\ Antti Karttunen, Jan 22 2020

a(n) = A000005(n) - A083910(n) - A083911(n) - A083913(n) - A083914(n) - A083915(n) - A083916(n) - A083917(n) - A083918(n) - A083919(n).
G.f.: Sum_{k>=1} x^(2*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(2,10) - (1 - gamma)/10 = 0.256367..., gamma(2,10) = -(psi(1/5) + log(10))/10 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Dec 30 2023

A083914 Number of divisors of n that are congruent to 4 modulo 10.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, May 08 2003

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
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, A200136 (psi(2/5)).
Cf. A083910, A083911, A083912, A083913, A083915, A083916, A083917, A083918, A083919.

Table[Count[Divisors[n],?(Mod[#,10]==4&)],{n,110}] (* _Harvey P. Dale, Dec 09 2014 *)
a[n_] := DivisorSum[n, 1 &, Mod[#, 10] == 4 &]; Array[a, 100] (* Amiram Eldar, Dec 30 2023 *)

A083914(n) = sumdiv(n,d,(4==(d%10))); \\ Antti Karttunen, Nov 07 2018

a(n) = A000005(n) - A083910(n) - A083911(n) - A083912(n) - A083913(n) - A083915(n) - A083916(n) - A083917(n) - A083918(n) - A083919(n).
G.f.: Sum_{k>=1} x^(4*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(4,10) - (1 - gamma)/10 = -0.0163984..., gamma(4,10) = -(psi(2/5) + log(10))/10 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Dec 30 2023

A083915 Number of divisors of n that are congruent to 5 modulo 10.

Original entry on oeis.org

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

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, A020759 (psi(1/2)).
Cf. A083910, A083911, A083912, A083913, A083914, A083916, A083917, A083918, A083919.

Table[Count[Divisors[n],?(Mod[#,10]==5&)],{n,120}] (* _Harvey P. Dale, Jan 26 2018 *)
a[n_] := DivisorSum[n, 1 &, Mod[#, 10] == 5 &]; Array[a, 100] (* Amiram Eldar, Dec 30 2023 *)

a(n) = sumdiv(n, d, d % 10 == 5); \\ Amiram Eldar, Dec 30 2023

a(n) = A000005(n) - A083910(n) - A083911(n) - A083912(n) - A083913(n) - A083914(n) - A083916(n) - A083917(n) - A083918(n) - A083919(n).
G.f.: Sum_{k>=1} x^(5*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(5,10) - (1 - gamma)/10 = 0.0761859..., gamma(5,10) = -(psi(1/2) + log(10))/10 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Dec 30 2023

A083916 Number of divisors of n that are congruent to 6 modulo 10.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 3, 0, 0, 0, 0, 0, 1, 0, 1, 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, A200137 (psi(3/5)).
Cf. A083910, A083911, A083912, A083913, A083914, A083915, A083917, A083918, A083919.

Table[Count[Divisors[n],?(Mod[#,10]==6&)],{n,110}] (* _Harvey P. Dale, Oct 16 2013 *)
a[n_] := DivisorSum[n, 1 &, Mod[#, 10] == 6 &]; Array[a, 100] (* Amiram Eldar, Dec 30 2023 *)

a(n) = sumdiv(n, d, d % 10 == 6); \\ Amiram Eldar, Dec 30 2023

a(n) = A000005(n) - A083910(n) - A083911(n) - A083912(n) - A083913(n) - A083914(n) - A083915(n) - A083917(n) - A083918(n) - A083919(n).
G.f.: Sum_{k>=1} x^(6*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(6,10) - (1 - gamma)/10 = -0.118475..., gamma(6,10) = -(psi(3/5) + log(10))/10 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Dec 30 2023

A083918 Number of divisors of n that are congruent to 8 modulo 10.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, May 08 2003

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
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, A200138 (psi(4/5)).
Cf. A083910, A083911, A083912, A083913, A083914, A083915, A083916, A083917, A083919.

Table[Count[Divisors[n],?(Mod[#,10]==8&)],{n,110}] (* _Harvey P. Dale, Sep 28 2016 *)
a[n_] := DivisorSum[n, 1 &, Mod[#, 10] == 8 &]; Array[a, 100] (* Amiram Eldar, Dec 30 2023 *)

a(n) = sumdiv(n, d, d % 10 == 8); \\ 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) - A083919(n).
G.f.: Sum_{k>=1} x^(8*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(8,10) - (1 - gamma)/10 = -0.176036..., gamma(8,10) = -(psi(4/5) + log(10))/10 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Dec 30 2023

A053511 Decimal expansion of log_10 (Pi).

Original entry on oeis.org

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

Offset: 0

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

			0.49714987269413385435126828829089887365167832438...

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A053510 (log(Pi)), A002392 (log(10)).

SetDefaultRealField(RealField(100)); R:= RealField(); Log(10, Pi(R)); // G. C. Greubel, May 15 2019

RealDigits[Log[10, Pi], 10, 105][[1]] (* Alonso del Arte, Sep 05 2012 *)

log(Pi)/log(10) \\ Charles R Greathouse IV, Sep 05 2012

numerical_approx(log(pi, 10), digits=100) # G. C. Greubel, May 15 2019

More terms from James Sellers, Jan 20 2000

A016643 Decimal expansion of log(20).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			2.995732273553990993435223576142540775676601622989028230154007910460966...

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Index entries for transcendental numbers

Cf. A016448 (continued fraction).

RealDigits[Log[20], 10, 150][[1]] (* Stefan Steinerberger, Apr 09 2006 *)

default(realprecision, 20080); x=log(20); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016643.txt", n, " ", d)); \\ Harry J. Smith, May 17 2009, corrected May 20 2009

Equals A002162 + A002392. - R. J. Mathar, Aug 13 2024

More terms from Stefan Steinerberger, Apr 09 2006

A016738 Continued fraction for log(10).

Original entry on oeis.org

2, 3, 3, 3, 1, 1, 3, 6, 3, 3, 1, 4, 2, 1, 2, 1, 3, 26, 5, 1, 23, 1, 1, 1, 2, 2, 3, 19, 1, 3, 716, 1, 2, 1, 1, 2, 2, 1, 22, 1, 17, 4, 1, 13, 7, 3, 5, 1, 1, 1, 1, 1, 7, 7, 1, 1, 4, 2, 1, 15, 1, 4, 2, 4, 1, 28, 1, 1, 2, 4, 1, 3, 6, 1, 9, 2, 4, 3, 1, 4, 3, 4, 1, 24, 16, 4, 6, 1, 33, 20, 9, 1, 1, 7, 24, 4, 2

Offset: 0

N. J. A. Sloane

			2.302585092994045684017991454... = 2 + 1/(3 + 1/(3 + 1/(3 + 1/(1 + ...)))). - _Harry J. Smith_, May 16 2009

Harry J. Smith, Table of n, a(n) for n = 0..19999
G. Xiao, Contfrac

Cf. A002392 (decimal expansion).

ContinuedFraction(Log(10)); // G. C. Greubel, Sep 15 2018

Mathematica
```
ContinuedFraction[ Log[10], 75 ]
```

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(log(10)); for (n=1, 20000, write("b016738.txt", n-1, " ", x[n])); } \\ Harry J. Smith, Apr 16 2009

More terms from Robert G. Wilson v, Dec 07 2000

Offset changed by Andrew Howroyd, Jul 10 2024

A059545 Beatty sequence for log(10).

Original entry on oeis.org

2, 4, 6, 9, 11, 13, 16, 18, 20, 23, 25, 27, 29, 32, 34, 36, 39, 41, 43, 46, 48, 50, 52, 55, 57, 59, 62, 64, 66, 69, 71, 73, 75, 78, 80, 82, 85, 87, 89, 92, 94, 96, 99, 101, 103, 105, 108, 110, 112, 115, 117, 119, 122, 124, 126, 128, 131, 133, 135, 138, 140, 142, 145, 147

Offset: 1

Mitch Harris, Jan 22 2001

Harry J. Smith, Table of n, a(n) for n = 1..2000
Aviezri S. Fraenkel, Jonathan Levitt, Michael Shimshoni, Characterization of the set of values f(n)=[n alpha], n=1,2,..., Discrete Math. 2 (1972), no.4, 335-345.
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences

Beatty complement is A059546.

Cf. A002392 (log(10)).

Table[Floor[Log[10]n], {n, 1, 100}] (* Stefan Steinerberger, May 09 2006 *)

{ default(realprecision, 100); b=log(10); for (n = 1, 2000, write("b059545.txt", n, " ", floor(n*b)); ) } \\ Harry J. Smith, Jun 28 2009

a(n) = floor(n*log(10)). - Stefan Steinerberger, May 09 2006

A059550 Beatty sequence for 1 + log(10).

Original entry on oeis.org

3, 6, 9, 13, 16, 19, 23, 26, 29, 33, 36, 39, 42, 46, 49, 52, 56, 59, 62, 66, 69, 72, 75, 79, 82, 85, 89, 92, 95, 99, 102, 105, 108, 112, 115, 118, 122, 125, 128, 132, 135, 138, 142, 145, 148, 151, 155, 158, 161, 165, 168, 171, 175, 178, 181, 184, 188, 191, 194, 198, 201, 204, 208, 211, 214, 217, 221, 224, 227, 231, 234, 237, 241, 244, 247, 250, 254, 257

Offset: 1

Mitch Harris, Jan 22 2001

Harry J. Smith, Table of n, a(n) for n = 1..2000
Aviezri S. Fraenkel, Jonathan Levitt and Michael Shimshoni, Characterization of the set of values f(n)=[n alpha], n=1,2,..., Discrete Math. 2 (1972), no.4, 335-345.
Eric Weisstein's World of Mathematics, Beatty Sequence
Index entries for sequences related to Beatty sequences

Beatty complement is A059549. Differs from A080081 at 76th term.

Cf. A002392.

Floor[Range[100]*(1 + Log[10])] (* Paolo Xausa, Jul 05 2024 *)

{ default(realprecision, 100); b=1 + log(10); for (n = 1, 2000, write("b059550.txt", n, " ", floor(n*b)); ) } \\ Harry J. Smith, Jun 28 2009

a(n) = floor(n*(1 + A002392)). - Paolo Xausa, Jul 05 2024

cp's OEIS Frontend

A083912 Number of divisors of n that are congruent to 2 modulo 10.

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A083914 Number of divisors of n that are congruent to 4 modulo 10.

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A083915 Number of divisors of n that are congruent to 5 modulo 10.

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A083916 Number of divisors of n that are congruent to 6 modulo 10.

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A083918 Number of divisors of n that are congruent to 8 modulo 10.

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A053511 Decimal expansion of log_10 (Pi).

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A016643 Decimal expansion of log(20).

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A016738 Continued fraction for log(10).

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