A020759 -id:A020759 - cp's OEIS frontend

A016627 Decimal expansion of log(4).

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

Offset: 1

N. J. A. Sloane

This constant (negated) is the 1-dimensional analog of Madelung's constant. - Jean-François Alcover, May 20 2014
This constant is the sum over the reciprocals of the hexagonal numbers A000384(n), n >= 1. See the Downey et al. link, and the formula by Robert G. Wilson v below. - Wolfdieter Lang, Sep 12 2016
log(4) - 1 is the mean ratio between the smaller length and the larger length of the two parts of a stick that is being broken at a point that is uniformly chosen at random (Mosteller, 1965). - Amiram Eldar, Jul 25 2020
From Bernard Schott, Sep 11 2020: (Start)
This constant was the subject of the problem B5 during the 42nd Putnam competition in 1981 (see formula Sep 11 2020 and Putnam link).
Jeffrey Shallit generalizes this result obtained for base 2 to any base b (see Amer. Math. Month. link): Sum_{k>=1} digsum(k)_b / (k*(k+1)) = (b/(b-1)) * log(b) where digsum(k)_b is the sum of the digits of k when expressed in base b (for base 10 see A334388). (End)

			1.38629436111989061883446424291635313615100026872051050824136...

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.
Frederick Mosteller, Fifty challenging problems of probability, Dover, New York, 1965. See problem 42, pp. 10 and 63.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 2, equation 2:13:8 at page 23.

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].
Lawrence Downey, Boon W. Ong, and James A. Sellers, Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, Coll. Math. J., 39, no. 5 (2008), 391-394.
I. S. Gradsteyn and I. M. Ryzhik, Table of integrals, series and products (6th ed.), 2000, (eq. 0.236.1 and 0.236.6).
D. H. Lehmer, Interesting series involving the Central Binomial Coefficient, Am. Math. Monthly, Vol. 92, No. 7 (1985) pp. 449-457.
Allon G. Percus, Gabriel Istrate, Bruno Goncalves, Robert Z. Sumi, and Stefan Boettcher, The Peculiar Phase Structure of Random Graph Bisection, arXiv:0808.1549 [cond-mat.stat-mech], 2008.
H.-J. Seiffert, Problem B-771, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 32, No. 4 (1994), p. 374; More Sums, Solution to Problem B-771 by Don Redmond, ibid., Vol. 33, No. 5 (1995), pp. 470-471.
J. O. Shallit, Solutions of Advanced Problems, 6450, The American Mathematical Monthly, Vol. 92, No. 7, Aug.-Sep., 1985, pp. 513-514; DOI: 10.2307/2322523.
42nd Putnam Competition, Problem B5, 1981.
Index entries for sequences related to Olympiads and other Mathematical competitions.
Index entries for transcendental numbers.

Cf. A016732 (continued fraction).
Cf. A002162 (half), A133362 (reciprocal).
Cf. A000384, A001787, A002378, A007531, A066066, A069072, A191907.
Cf. A000045, A000120, A334388.

RealDigits[Log@ 4, 10, 111][[1]] (* Robert G. Wilson v, Aug 31 2014 *)

default(realprecision, 20080); x=log(4); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016627.txt", n, " ", d)); \\ Harry J. Smith, May 16 2009, corrected May 19 2009

A016627_vec(N)=digits(floor(log(precision(4.,N))*10^(N-1))) \\ Or: default(realprecision,N);digits(log(4)\.1^N) \\ M. F. Hasler, Oct 20 2013

log(4) = Sum_{k >= 1} H(k)/2^k where H(k) is the k-th harmonic number. - Benoit Cloitre, Jun 15 2003
Equals 1 - Sum_{k >= 1} (-1)^k/A002378(k) = 1 + 2*Sum_{k >= 0} 1/A069072(k) = 5/4 - Sum_{k >= 1} (-1)^k/A007531(k+2). - R. J. Mathar, Jan 23 2009
Equals 2*A002162 = Sum_{n >= 1} binomial(2*n, n)/(n*4^n) [D. H. Lehmer, Am. Math. Monthly 92 (1985) 449 and Jolley eq. 262]. - R. J. Mathar, Mar 04 2009
log(4) = Sum_{k >= 1} A191907(4, k)/k, (conjecture). - Mats Granvik, Jun 19 2011
log(4) = lim_{n -> infinity} A066066(n)/n. - M. F. Hasler, Oct 20 2013
Equals Sum_{k >= 1} 1/( 2*k^2 - k ). - Robert G. Wilson v, Aug 31 2014
Equals gamma(0, 1/2) - gamma(0, 1) = -(EulerGamma + polygamma(0, 1/2)), where gamma(n,x) denotes the generalized Stieltjes constants, see A020759. - Peter Luschny, May 16 2018
From Amiram Eldar, Jul 25 2020: (Start)
Equals Sum_{k>=1} (3/4)^k/k.
Equals Sum_{k>=1} 1/(k*2^(k-1)) = Sum_{k>=1} 1/A001787(k).
Equals Integral_{x=0..1} log(1+1/x) dx. (End)
Equals Sum_{k>=1} A000120(k) / (k*(k+1)). - Bernard Schott, Sep 11 2020
Equals 1 + Sum_{k>=1} zeta(2*k+1)/4^k. - Amiram Eldar, May 27 2021
Equals Sum_{k>=1} (2*k+1)*Fibonacci(k)/(k*(k+1)*2^k) (Seiffert, 1994). - Amiram Eldar, Jan 15 2022
Continued fraction: log(4) = 1 + 1/(2 + (1*2)/(2 + (2*3)/(2 + (3*4)/(2 + (4*5)/(2 + ... ))))) due to Euler. - Peter Bala, Mar 05 2024
log(4) = 2*Sum_{k>=1} 1/(k*P(k, 5/3)*P(k-1, 5/3)), where P(k, x) denotes the k-th Legendre polynomial. The first 20 terms of the series gives log(4) correct to 18 decimal places. - Peter Bala, Mar 18 2024
Equals Sum_{k>=1} (2*k - 1)!!/(k*(2*k)!!) [Ross] (see Spanier at p. 23). - Stefano Spezia, Dec 27 2024
Equals 1 + Sum_{k>=1} 1/(k*(4*k^2-1)). - Sean A. Irvine, Apr 05 2025
Equals Sum_{k>=1} (12*k^2-1)/(k*(4*k^2-1)^2). - Sean A. Irvine, Apr 06 2025
Equals Integral_{x=0..1} arctanh(sqrt(x))/sqrt(x) dx. - Kritsada Moomuang, Jun 06 2025
From Kritsada Moomuang, Jun 18 2025: (Start)
Equals Integral_{x=0..1} (x^(n - 1)*(x^(3*n) - 1))/log(x) dx, for n > 0.
Equals Integral_{x=0..Pi} sin(x)/(1 + abs(cos(x))) dx. (End)

A228725 Decimal expansion of the generalized Euler constant gamma(1,2).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Aug 31 2013

The complement (A239097) is gamma(0,2) = lim_{x->oo} ((Sum_{n=1..x, n even} 1/n) - log(x)/2) = (A001620 - A002162)/2 = -0.05796575... - R. J. Mathar, Sep 06 2013

			0.63518142273073908501187210577028949955882973515008942646322...

G. C. Greubel, Table of n, a(n) for n = 0..10000
M. L. Glasser, A note on Beukers's and related integrals, Amer. Math. Monthly 126(4) (2019), 361-363.
J. C. Lagarias, Euler's constant: Euler's work and modern developments, arXiv:1303.1856 [math.NT], 2013. See Section 3.8.
D. H. Lehmer, Euler constants for arithmetic progressions, Acta Arith. 27 (1975), 125-142.

Cf. A001620, A002162, A020759, A239097.

SetDefaultRealField(RealField(100)); R:= RealField();
(EulerGamma + Log(2))/2; // G. C. Greubel, Aug 27 2018

Maple
```
(gamma+log(2))/2 ; evalf(%) ;
```

RealDigits[(EulerGamma+Log[2])/2,10,120][[1]] (* Harvey P. Dale, Dec 26 2013 *)

(Euler+log(2))/2 \\ Charles R Greathouse IV, Jul 21 2015

Equals lim_{x->oo} ((Sum_{n=1..x, n odd} 1/n) - log(x)/2).
Equals (A001620 + A002162)/2.
From Amiram Eldar, Jun 30 2020: (Start)
Equals -Integral_{x=0..1} log(log(1/x))*x dx.
Equals -Integral_{x=0..oo} exp(-2*x)*log(x) dx. (End)
Equals Integral_{x=0..1, y=0..1} log(-log(x*y))*x*y/log(x*y) dx dy. (Apply Theorem 1 or Theorem 2 of Glasser (2019) to one of Amiram Eldar's integrals.) - Petros Hadjicostas, Jun 30 2020
Equals -(psi(1/2) + log(2))/2 = (A020759 - A002162)/2. - Amiram Eldar, Jan 07 2024

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

A200134 Decimal expansion of the negated value of the digamma function at 3/4.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Nov 13 2011

			Psi(3/4) = -1.085860879786472169626886762817...

G. C. Greubel, Table of n, a(n) for n = 1..10000
E. D. Krupnikov, K. S. Kölbig, Some special cases of the generalized hypergeometric function (q+1)Fq, J. Comp. Appl. Math. 78 (1997) 79-95.
Wikipedia, Digamma function
Index entries for sequences related to the digamma function

Cf. A001620, A020759, A020777, A047787, A301816.

SetDefaultRealField(RealField(100)); R:= RealField(); -EulerGamma(R) + Pi(R)/2 - 3*Log(2); // G. C. Greubel, Aug 29 2018

Maple
```
evalf(-gamma+Pi/2-3*log(2)) ;
```

RealDigits[ -PolyGamma[3/4], 10, 97] // First (* Jean-François Alcover, Feb 20 2013 *)
N[StieltjesGamma[0, 3/4], 99] (* Peter Luschny, May 16 2018 *)

-psi(3/4) \\ Charles R Greathouse IV, Nov 22 2011

Psi(3/4) = -gamma + Pi/2 - 3*log(2) = A000796 - A020777 = 3.14159... - 4.22745...

Pi = gamma(0,1/4) - gamma(0,3/4) = A020777 - A200134, where gamma(n,x) denotes the generalized Stieltjes constants. - Peter Luschny, May 16 2018

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

A200064 Decimal expansion of the negated value of the digamma function at 2/3.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Nov 13 2011

			psi(2/3) = -1.3182344157865884724023408166...

Cf. A020759, A047787, A020777

RealDigits[ -PolyGamma[2/3], 10, 98] // First (* Jean-François Alcover, Feb 20 2013 *)

-psi(2/3) \\ Charles R Greathouse IV, Nov 22 2011

psi(2/3) = -gamma+Pi*sqrt(3)/6-3*log(3)/2 = A093602 - A047787 = 1.813799 -3.132033...

A222457 Decimal expansion of the negated value of the digamma function at 1/6.

Original entry on oeis.org

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

Offset: 1

Bruno Berselli, Feb 21 2013

			Psi(1/6) = -6.3321275053749147924249615748457777225904948...

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Ernst D. Krupnikov and K. S. Kolbig, Some special cases of the generalized hypergeometric function _{q+1}F_q, J. Comp. Appl. Math. 78 (1997) 79-95
Eric Weisstein's World of Mathematics, Gauss's Digamma Theorem.
Wikipedia, Digamma function: special values.
Index entries for sequences related to the digamma function

Cf. A020759, A047787, A200064, A222458.

RealDigits[-PolyGamma[1/6], 10, 90][[1]]

Maxima
```
fpprec:90; ev(bfloat(-psi[0](1/6)));
```
PARI
```
-psi(1/6)
```

Psi(1/6) = -gamma -Pi*sqrt(3)/2 -3*log(3)/2 -2*log(2).

A200138 Decimal expansion of the negated value of the digamma function at 4/5.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Nov 13 2011

			Psi(4/5) = -0.965008566706138459391297633...

Index entries for sequences related to the digamma function

Cf. A020759, A047787, A020777, A200134-A200137.

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

RealDigits[ -PolyGamma[4/5], 10, 87] // First (* Jean-François Alcover, Feb 20 2013 *)

-psi(4/5) \\ Charles R Greathouse IV, Nov 22 2011

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

A222458 Decimal expansion of the negated value of the digamma function at 5/6.

Original entry on oeis.org

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

Offset: 0

Bruno Berselli, Feb 21 2013

			Psi(5/6) = -0.890729412672261240642726801919310525738296...

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Eric Weisstein's World of Mathematics, Gauss's Digamma Theorem.
Wikipedia, Digamma function: Gauss's digamma theorem.
Index entries for sequences related to the digamma function

Cf. A020759, A047787, A200064, A222457.

RealDigits[-PolyGamma[5/6], 10, 90][[1]]

Maxima
```
fpprec:90; ev(bfloat(-psi[0](5/6)));
```
PARI
```
-psi(5/6)
```

Psi(5/6) = -gamma + Pi*sqrt(3)/2 - 3*log(3)/2 - 2*log(2).

A320003 Number of proper divisors of n of the form 6*k + 3.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 03 2018

Number of divisors of n that are odd multiples of 3 and less than n.

			For n = 18, of its five proper divisors [1, 2, 3, 6, 9] only 3 and 9 are odd multiples of three, thus a(18) = 2.
For n = 108, the odd part is 27 for which 27/3 has 3 divisors. As 108 is even, we don't subtract 1 from that 3 to get a(108) = 3. - _David A. Corneth_, Oct 03 2018

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. A319990, A320001, A320005.

Cf. A001620, A016629, A020759 (psi(1/2)).

a[n_] := DivisorSum[n, 1 &, # < n && Mod[#, 6] == 3 &]; Array[a, 100] (* Amiram Eldar, Nov 25 2023 *)

A320003(n) = if(!n,n,sumdiv(n, d, (d

a(n) = if(n%3==0, my(v=valuation(n, 2)); n>>=v; numdiv(n/3)-(!v), 0) \\ David A. Corneth, Oct 03 2018

a(n) = Sum_{d|n, dA000035(d))*A079978(d).
a(n) = A007814(A319990(n)).
a(4*n) = a(2*n). - David A. Corneth, Oct 03 2018
G.f.: Sum_{k>=1} x^(12*k-6) / (1 - x^(6*k-3)). - Ilya Gutkovskiy, Apr 14 2021
Sum_{k=1..n} a(k) = n*log(n)/6 + c*n + O(n^(1/3)*log(n)), where c = gamma(3,6) - (2 - gamma)/6 = -0.208505..., gamma(3,6) = -(psi(1/2) + log(6))/6 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

cp's OEIS Frontend

A016627 Decimal expansion of log(4).

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A228725 Decimal expansion of the generalized Euler constant gamma(1,2).

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

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A200134 Decimal expansion of the negated value of the digamma function at 3/4.

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

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PARI

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A200064 Decimal expansion of the negated value of the digamma function at 2/3.

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Mathematica

PARI

Formula

A222457 Decimal expansion of the negated value of the digamma function at 1/6.

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