A200138 -id:A200138 - cp's OEIS frontend

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

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

A200135 Decimal expansion of the negated value of the digamma function at 1/5.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Nov 13 2011

			Psi(1/5) =  -5.289039896592188295547207962...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Wikipedia, Digamma function
Index entries for sequences related to the digamma function

Cf. A020759, A047787, A020777, A200064, A200134, A200136, A200137, A200138.

SetDefaultRealField(RealField(100)); R:= RealField(); -EulerGamma(R) -Pi(R)*Sqrt(1+2/Sqrt(5))/2 -5*Log(5)/4 -Sqrt(5)/4*Log((3+Sqrt(5)/2) ); // G. C. Greubel, Sep 03 2018

-gamma-Pi*sqrt(1+2/sqrt(5))/2-5*log(5)/4-sqrt(5)/4*log((3+sqrt(5)/2) ); evalf(%) ;

RealDigits[-PolyGamma[1/5], 10, 105] // First (* Jean-François Alcover, Feb 11 2013 *)

-psi(1/5) \\ Charles R Greathouse IV, Jul 19 2013

Psi(1/5) = -gamma - Pi*sqrt(1 + 2/sqrt(5))/2 - 5*log(5)/4 -sqrt(5)*log((3 + sqrt(5))/2)/4 where gamma = A001620, sqrt(1 + 2/sqrt(5)) = A019952, (3 + sqrt(5))/2 = A104457.

More terms from Jean-François Alcover, Feb 11 2013

A200136 Decimal expansion of the negated value of the digamma function at 2/5.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Nov 13 2011

			Psi(2/5) = -2.5613845445851161457306754820475...

Index entries for sequences related to the digamma function

Cf. A020759, A020777, A047787, A200064, A200135-A200138.

-gamma-Pi*sqrt(1-2/sqrt(5))/2-5*log(5)/4+sqrt(5)*log((3+sqrt(5))/2)/4 ; evalf(%) ;

RealDigits[ PolyGamma[2/5], 10, 84] // First (* Jean-François Alcover, Feb 21 2013 *)

-psi(2/5) \\ Charles R Greathouse IV, Jul 19 2013

Psi(2/5) = -gamma -Pi*sqrt(1-2/sqrt 5)/2 -5*log(5)/4 +sqrt(5)*log((3+sqrt 5)/2)/4.

A200137 Decimal expansion of the negated digamma function at 3/5.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Nov 13 2011

			Psi(3/5) = -1.540619213893190414760663948806231941510...

Index entries for sequences related to the digamma function

Cf. A020759, A047787, A020777, A200134-A200138

-gamma+Pi*sqrt(1-2/sqrt(5))/2-5*log(5)/4+sqrt(5)/4*log(3/2+sqrt(5)/2) ; evalf(%) ;

RealDigits[-PolyGamma[3/5], 10, 105] // First (* Jean-François Alcover, Feb 11 2013 *)

-psi(3/5) \\ Charles R Greathouse IV, Jul 19 2013

Psi(3/5) = -gamma +Pi*sqrt( 1-2/sqrt 5)/2 -5*log(5)/4 +sqrt(5)*log((3+sqrt 5)/2)/4.

More terms from Jean-François Alcover, Feb 11 2013

A001545 a(n) = (5n+1)(5*n+4).

Original entry on oeis.org

4, 54, 154, 304, 504, 754, 1054, 1404, 1804, 2254, 2754, 3304, 3904, 4554, 5254, 6004, 6804, 7654, 8554, 9504, 10504, 11554, 12654, 13804, 15004, 16254, 17554, 18904, 20304, 21754, 23254, 24804, 26404, 28054, 29754, 31504, 33304, 35154, 37054, 39004, 41004, 43054

Offset: 0

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A001622, A016861, A016897, A177059, A200135, A200138.

Table[(5n+1)(5n+4),{n,0,60}] (* or *) LinearRecurrence[{3,-3,1},{4,54,154},60] (* Harvey P. Dale, Mar 17 2019 *)

a(n)=(5*n+1)*(5*n+4) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 50*A000217(n) + 4.
a(n) = 50*n + a(n-1) with a(0)=4. - Vincenzo Librandi, Jan 20 2011
From Amiram Eldar, Jan 23 2022: (Start)
Sum_{n>=0} 1/a(n) = sqrt(1 + 2/sqrt(5))*Pi/15 = 0.2882687....
Sum_{n>=0} (-1)^n/a(n) = 2*log(phi)/(3*sqrt(5)) + 2*log(2)/15, where phi is the golden ratio (A001622).
Product_{n>=0} (1 - 1/a(n)) = sqrt(2 + 2/sqrt(5)) * cos(sqrt(13)*Pi/10).
Product_{n>=0} (1 + 1/a(n)) = sqrt(2 + 2/sqrt(5)) * cos(Pi/(2*sqrt(5))).
Product_{n>=0} (1 + 2/a(n)) = phi. (End)
G.f.: 2*(2+21*x+2*x^2)/(1-x)^3. - R. J. Mathar, May 30 2022
Sum_{n>=0} 1/a(n) = (Psi(4/5) - Psi(1/5))/15. See A200135, A200138. - R. J. Mathar, May 30 2022
From Elmo R. Oliveira, Oct 23 2024: (Start)
E.g.f.: exp(x)*(4 + 25*x*(2 + x)).
a(n) = A016861(n)*A016897(n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

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

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A200135 Decimal expansion of the negated value of the digamma function at 1/5.

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A200136 Decimal expansion of the negated value of the digamma function at 2/5.

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A200137 Decimal expansion of the negated digamma function at 3/5.

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A001545 a(n) = (5*n+1)*(5*n+4).

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A001545 a(n) = (5n+1)(5*n+4).